Functions that compute some of the sequences in Sloane’s tables - Combinatorics

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# Functions that compute some of the sequences in Sloane’s tables

EXAMPLES:

Type `sloane.[tab]` to see a list of the sequences that are defined.

Sage

    sage: a = sloane.A000005; a
    The integer sequence tau(n), which is the number of divisors of n.
    sage: a(1)
    1
    sage: a(6)
    4
    sage: a(100)
    9

Python

    >>> from sage.all import *
    >>> a = sloane.A000005; a
    The integer sequence tau(n), which is the number of divisors of n.
    >>> a(Integer(1))
    1
    >>> a(Integer(6))
    4
    >>> a(Integer(100))
    9

Type `d._eval??` to see how the function that computes an individual term of the sequence is implemented.

The input must be a positive integer:

Sage

    sage: a(0)
    Traceback (most recent call last):
    ...
    ValueError: input n (=0) must be a positive integer
    sage: a(1/3)
    Traceback (most recent call last):
    ...
    TypeError: input must be an int or Integer

[Python]

    >>> from sage.all import *
    >>> a(Integer(0))
    Traceback (most recent call last):
    ...
    ValueError: input n (=0) must be a positive integer
    >>> a(Integer(1)/Integer(3))
    Traceback (most recent call last):
    ...
    TypeError: input must be an int or Integer

You can also change how a sequence prints:

Sage

    sage: a = sloane.A000005; a
    The integer sequence tau(n), which is the number of divisors of n.
    sage: a.rename('(..., tau(n), ...)')
    sage: a
    (..., tau(n), ...)
    sage: a.reset_name()
    sage: a
    The integer sequence tau(n), which is the number of divisors of n.

Python

    >>> from sage.all import *
    >>> a = sloane.A000005; a
    The integer sequence tau(n), which is the number of divisors of n.
    >>> a.rename('(..., tau(n), ...)')
    >>> a
    (..., tau(n), ...)
    >>> a.reset_name()
    >>> a
    The integer sequence tau(n), which is the number of divisors of n.

See also

- If you want to get more informations relative to a sequence (references, links, examples, programs, …), you can use the On-Line Encyclopedia of Integer Sequences provided by the `OEIS` module.

- If you plan to do a lot of automatic searches for subsequences, you should consider installing `SloaneEncyclopedia`, a local partial copy of the OEIS.

AUTHORS:

- William Stein: framework

- Jaap Spies: most sequences

- Nick Alexander: updated framework

class sage.combinat.sloane_functions.A000001

  Bases: `SloaneSequence`

  Number of groups of order \(n\).

  INPUT:

  - `n` – positive integer

  OUTPUT: integer

  EXAMPLES:

  Sage

      sage: a = sloane.A000001;a
      Number of groups of order n.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      1
      sage: a(9)
      2
      sage: a.list(16)
      [1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14]
      sage: a(60)
      13

  Python

      >>> from sage.all import *
      >>> a = sloane.A000001;a
      Number of groups of order n.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      1
      >>> a(Integer(9))
      2
      >>> a.list(Integer(16))
      [1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14]
      >>> a(Integer(60))
      13

  AUTHORS:

  - Jaap Spies (2007-02-04)

class sage.combinat.sloane_functions.A000004

  Bases: `SloaneSequence`

  The zero sequence.

  INPUT:

  - `n` – nonnegative integer

  EXAMPLES:

  Sage

      sage: a = sloane.A000004; a
      The zero sequence.
      sage: a(1)
      0
      sage: a(2007)
      0
      sage: a.list(12)
      [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000004; a
      The zero sequence.
      >>> a(Integer(1))
      0
      >>> a(Integer(2007))
      0
      >>> a.list(Integer(12))
      [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

  AUTHORS:

  - Jaap Spies (2006-12-10)

class sage.combinat.sloane_functions.A000005

  Bases: `SloaneSequence`

  The sequence \(tau(n)\), which is the number of divisors of \(n\).

  This sequence is also denoted \(d(n)\) (also called \(\tau(n)\) or \(\sigma_0(n)\)), the number of divisors of \(n\).

  INPUT:

  - `n` – positive integer

  EXAMPLES:

  Sage

      sage: d = sloane.A000005; d
      The integer sequence tau(n), which is the number of divisors of n.
      sage: d(1)
      1
      sage: d(6)
      4
      sage: d(51)
      4
      sage: d(100)
      9
      sage: d(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: d.list(10)
      [1, 2, 2, 3, 2, 4, 2, 4, 3, 4]

  Python

      >>> from sage.all import *
      >>> d = sloane.A000005; d
      The integer sequence tau(n), which is the number of divisors of n.
      >>> d(Integer(1))
      1
      >>> d(Integer(6))
      4
      >>> d(Integer(51))
      4
      >>> d(Integer(100))
      9
      >>> d(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> d.list(Integer(10))
      [1, 2, 2, 3, 2, 4, 2, 4, 3, 4]

  AUTHORS:

  - Jaap Spies (2006-12-10)

  - William Stein (2007-01-08)

class sage.combinat.sloane_functions.A000007

  Bases: `SloaneSequence`

  The characteristic function of 0: \(a(n) = 0^n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000007;a
      The characteristic function of 0: a(n) = 0^n.
      sage: a(0)
      1
      sage: a(2)
      0
      sage: a(12)
      0
      sage: a.list(12)
      [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000007;a
      The characteristic function of 0: a(n) = 0^n.
      >>> a(Integer(0))
      1
      >>> a(Integer(2))
      0
      >>> a(Integer(12))
      0
      >>> a.list(Integer(12))
      [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

  AUTHORS:

  - Jaap Spies (2007-01-12)

class sage.combinat.sloane_functions.A000008

  Bases: `SloaneSequence`

  Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000008;a
      Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(13)
      16
      sage: a.list(14)
      [1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000008;a
      Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      1
      >>> a(Integer(13))
      16
      >>> a.list(Integer(14))
      [1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16]

  AUTHOR:

  - 10. Gaski (2009-05-29)

class sage.combinat.sloane_functions.A000009

  Bases: `SloaneSequence`

  Number of partitions of \(n\) into odd parts.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000009;a
      Number of partitions of n into odd parts.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(13)
      18
      sage: a.list(14)
      [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000009;a
      Number of partitions of n into odd parts.
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      1
      >>> a(Integer(13))
      18
      >>> a.list(Integer(14))
      [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

  AUTHOR:

  - Jaap Spies (2007-01-30)

  cf()

    EXAMPLES:

    Sage

        sage: it = sloane.A000009.cf()
        sage: [next(it) for i in range(14)]
        [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

    Python

        >>> from sage.all import *
        >>> it = sloane.A000009.cf()
        >>> [next(it) for i in range(Integer(14))]
        [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A000009.list(14)
        [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

    Python

        >>> from sage.all import *
        >>> sloane.A000009.list(Integer(14))
        [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

class sage.combinat.sloane_functions.A000010

  Bases: `SloaneSequence`

  The integer sequence A000010 is Euler’s totient function.

  Number of positive integers \(i < n\) that are relative prime to \(n\). Number of totatives of \(n\).

  Euler totient function \(\phi(n)\): count numbers \(n\) and prime to \(n\). `euler_phi` is a standard Sage function implemented in PARI.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000010; a
      Euler's totient function
      sage: a(1)
      1
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(11)
      10
      sage: a.list(12)
      [1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4]
      sage: a(1/3)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  Python

      >>> from sage.all import *
      >>> a = sloane.A000010; a
      Euler's totient function
      >>> a(Integer(1))
      1
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(11))
      10
      >>> a.list(Integer(12))
      [1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4]
      >>> a(Integer(1)/Integer(3))
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  AUTHORS:

  - Jaap Spies (2007-01-12)

class sage.combinat.sloane_functions.A000012

  Bases: `SloaneSequence`

  The all 1s sequence.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000012; a
      The all 1s sequence.
      sage: a(1)
      1
      sage: a(2007)
      1
      sage: a.list(12)
      [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000012; a
      The all 1s sequence.
      >>> a(Integer(1))
      1
      >>> a(Integer(2007))
      1
      >>> a.list(Integer(12))
      [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

  AUTHORS:

  - Jaap Spies (2007-01-12)

class sage.combinat.sloane_functions.A000015

  Bases: `SloaneSequence`

  Smallest prime power \(\geq n\) (where \(1\) is considered a prime power).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000015; a
      Smallest prime power >= n.
      sage: a(1)
      1
      sage: a(8)
      8
      sage: a(305)
      307
      sage: a(-4)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be a positive integer
      sage: a.list(12)
      [1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 13]
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  Python

      >>> from sage.all import *
      >>> a = sloane.A000015; a
      Smallest prime power >= n.
      >>> a(Integer(1))
      1
      >>> a(Integer(8))
      8
      >>> a(Integer(305))
      307
      >>> a(-Integer(4))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be a positive integer
      >>> a.list(Integer(12))
      [1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 13]
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  AUTHORS:

  - Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000016

  Bases: `SloaneSequence`

  Sloane’s A000016.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000016; a
      Sloane's A000016.
      sage: a(1)
      1
      sage: a(0)
      1
      sage: a(8)
      16
      sage: a(75)
      251859545753048193000
      sage: a(-4)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be an integer >= 0
      sage: a.list(12)
      [1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000016; a
      Sloane's A000016.
      >>> a(Integer(1))
      1
      >>> a(Integer(0))
      1
      >>> a(Integer(8))
      16
      >>> a(Integer(75))
      251859545753048193000
      >>> a(-Integer(4))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be an integer >= 0
      >>> a.list(Integer(12))
      [1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94]

  AUTHORS:

  - Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000027

  Bases: `SloaneSequence`

  The natural numbers. Also called the whole numbers, the counting numbers or the positive integers.

  The following examples are tests of SloaneSequence more than A000027.

  EXAMPLES:

  Sage

      sage: s = sloane.A000027; s
      The natural numbers.
      sage: s(10)
      10

  Python

      >>> from sage.all import *
      >>> s = sloane.A000027; s
      The natural numbers.
      >>> s(Integer(10))
      10

  Index n is interpreted as _eval(n):

  Sage

      sage: s[10]
      10

  [Python]

      >>> from sage.all import *
      >>> s[Integer(10)]
      10

  Slices are interpreted with absolute offsets, so the following returns the terms of the sequence up to but not including the third term:

  Sage

      sage: s[:3]
      [1, 2]
      sage: s[3:6]
      [3, 4, 5]
      sage: s.list(5)
      [1, 2, 3, 4, 5]

  Python

      >>> from sage.all import *
      >>> s[:Integer(3)]
      [1, 2]
      >>> s[Integer(3):Integer(6)]
      [3, 4, 5]
      >>> s.list(Integer(5))
      [1, 2, 3, 4, 5]

  link = 'http://oeis.org/classic/A000027'

class sage.combinat.sloane_functions.A000030

  Bases: `SloaneSequence`

  Initial digit of \(n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000030; a
      Initial digit of n
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(8)
      8
      sage: a(454)
      4
      sage: a(-4)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be an integer >= 0
      sage: a.list(12)
      [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000030; a
      Initial digit of n
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(8))
      8
      >>> a(Integer(454))
      4
      >>> a(-Integer(4))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be an integer >= 0
      >>> a.list(Integer(12))
      [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1]

  AUTHORS:

  - Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000032

  Bases: `SloaneSequence`

  Lucas numbers (beginning at 2): \(L(n) = L(n-1) + L(n-2)\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000032; a
      Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).
      sage: a(0)
      2
      sage: a(1)
      1
      sage: a(8)
      47
      sage: a(200)
      627376215338105766356982006981782561278127
      sage: a(-4)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be an integer >= 0
      sage: a.list(12)
      [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000032; a
      Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).
      >>> a(Integer(0))
      2
      >>> a(Integer(1))
      1
      >>> a(Integer(8))
      47
      >>> a(Integer(200))
      627376215338105766356982006981782561278127
      >>> a(-Integer(4))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be an integer >= 0
      >>> a.list(Integer(12))
      [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199]

  AUTHORS:

  - Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000035

  Bases: `SloaneSequence`

  A simple periodic sequence.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000035;a
      A simple periodic sequence.
      sage: a(0.0)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer
      sage: a(1)
      1
      sage: a(2)
      0
      sage: a(9)
      1
      sage: a.list(10)
      [0, 1, 0, 1, 0, 1, 0, 1, 0, 1]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000035;a
      A simple periodic sequence.
      >>> a(RealNumber('0.0'))
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      0
      >>> a(Integer(9))
      1
      >>> a.list(Integer(10))
      [0, 1, 0, 1, 0, 1, 0, 1, 0, 1]

  AUTHORS:

  - Jaap Spies (2007-02-02)

class sage.combinat.sloane_functions.A000040

  Bases: `SloaneSequence`

  The prime numbers.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000040; a
      The prime numbers.
      sage: a(1)
      2
      sage: a(8)
      19
      sage: a(305)
      2011
      sage: a.list(12)
      [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  Python

      >>> from sage.all import *
      >>> a = sloane.A000040; a
      The prime numbers.
      >>> a(Integer(1))
      2
      >>> a(Integer(8))
      19
      >>> a(Integer(305))
      2011
      >>> a.list(Integer(12))
      [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  AUTHORS:

  - Jaap Spies (2007-01-17)

class sage.combinat.sloane_functions.A000041

  Bases: `SloaneSequence`

  \(a(n)\) = number of partitions of \(n\) (the partition numbers).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000041;a
      a(n) = number of partitions of n (the partition numbers).
      sage: a(0)
      1
      sage: a(2)
      2
      sage: a(8)
      22
      sage: a(200)
      3972999029388
      sage: a.list(9)
      [1, 1, 2, 3, 5, 7, 11, 15, 22]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000041;a
      a(n) = number of partitions of n (the partition numbers).
      >>> a(Integer(0))
      1
      >>> a(Integer(2))
      2
      >>> a(Integer(8))
      22
      >>> a(Integer(200))
      3972999029388
      >>> a.list(Integer(9))
      [1, 1, 2, 3, 5, 7, 11, 15, 22]

  AUTHORS:

  - Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000043

  Bases: `SloaneSequence`

  Primes \(p\) such that \(2^p - 1\) is prime. \(2^p - 1\) is then called a Mersenne prime.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000043;a
      Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime.
      sage: a(1)
      2
      sage: a(2)
      3
      sage: a(39)
      13466917
      sage: a(40)
      Traceback (most recent call last):
      ...
      IndexError: list index out of range
      sage: a.list(12)
      [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000043;a
      Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime.
      >>> a(Integer(1))
      2
      >>> a(Integer(2))
      3
      >>> a(Integer(39))
      13466917
      >>> a(Integer(40))
      Traceback (most recent call last):
      ...
      IndexError: list index out of range
      >>> a.list(Integer(12))
      [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000045

  Bases: `SloaneSequence`

  Sequence of Fibonacci numbers, offset 0,4.

  REFERENCES:

  - S. Plouffe, Project Gutenberg, The First 1001 Fibonacci Numbers, http://ibiblio.org/pub/docs/books/gutenberg/etext01/fbncc10.txt

  We have one more. Our first Fibonacci number is 0.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000045; a
      Fibonacci numbers with index n >= 0
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a.list(12)
      [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
      sage: a(1/3)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  Python

      >>> from sage.all import *
      >>> a = sloane.A000045; a
      Fibonacci numbers with index n >= 0
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a.list(Integer(12))
      [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
      >>> a(Integer(1)/Integer(3))
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  AUTHORS:

  - Jaap Spies (2007-01-13)

  fib()

    Return a generator over all Fibonacci numbers, starting with 0.

    EXAMPLES:

    Sage

        sage: it = sloane.A000045.fib()
        sage: [next(it) for i in range(10)]
        [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

    Python

        >>> from sage.all import *
        >>> it = sloane.A000045.fib()
        >>> [next(it) for i in range(Integer(10))]
        [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A000045.list(10)
        [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

    Python

        >>> from sage.all import *
        >>> sloane.A000045.list(Integer(10))
        [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

class sage.combinat.sloane_functions.A000069

  Bases: `SloaneSequence`

  Odious numbers: odd number of 1s in binary expansion.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000069; a
      Odious numbers: odd number of 1s in binary expansion.
      sage: a(0)
      1
      sage: a(2)
      4
      sage: a.list(9)
      [1, 2, 4, 7, 8, 11, 13, 14, 16]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000069; a
      Odious numbers: odd number of 1s in binary expansion.
      >>> a(Integer(0))
      1
      >>> a(Integer(2))
      4
      >>> a.list(Integer(9))
      [1, 2, 4, 7, 8, 11, 13, 14, 16]

  AUTHORS:

  - Jaap Spies (2007-02-02)

class sage.combinat.sloane_functions.A000073

  Bases: `SloaneSequence`

  Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 0, 0, 1, …

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000073;a
      Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
      sage: a(0)
      0
      sage: a(1)
      0
      sage: a(2)
      1
      sage: a(11)
      149
      sage: a.list(12)
      [0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000073;a
      Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      0
      >>> a(Integer(2))
      1
      >>> a(Integer(11))
      149
      >>> a.list(Integer(12))
      [0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149]

  AUTHORS:

  - Jaap Spies (2007-01-19)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A000073.list(10)
        [0, 0, 1, 1, 2, 4, 7, 13, 24, 44]

    Python

        >>> from sage.all import *
        >>> sloane.A000073.list(Integer(10))
        [0, 0, 1, 1, 2, 4, 7, 13, 24, 44]

class sage.combinat.sloane_functions.A000079

  Bases: `SloaneSequence`

  Powers of 2: \(a(n) = 2^n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000079;a
      Powers of 2: a(n) = 2^n.
      sage: a(0)
      1
      sage: a(2)
      4
      sage: a(8)
      256
      sage: a(100)
      1267650600228229401496703205376
      sage: a.list(9)
      [1, 2, 4, 8, 16, 32, 64, 128, 256]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000079;a
      Powers of 2: a(n) = 2^n.
      >>> a(Integer(0))
      1
      >>> a(Integer(2))
      4
      >>> a(Integer(8))
      256
      >>> a(Integer(100))
      1267650600228229401496703205376
      >>> a.list(Integer(9))
      [1, 2, 4, 8, 16, 32, 64, 128, 256]

  AUTHORS:

  - Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000085

  Bases: `SloaneSequence`

  Number of self-inverse permutations on \(n\) letters, also known as involutions; number of Young tableaux with \(n\) cells.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000085;a
      Number of self-inverse permutations on n letters.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(12)
      140152
      sage: a.list(13)
      [1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000085;a
      Number of self-inverse permutations on n letters.
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      2
      >>> a(Integer(12))
      140152
      >>> a.list(Integer(13))
      [1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152]

  AUTHORS:

  - Jaap Spies (2007-02-03)

class sage.combinat.sloane_functions.A000100

  Bases: `SloaneSequence`

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000100;a
      Number of compositions of n in which the maximum part size is 3.
      sage: a(0)
      0
      sage: a(1)
      0
      sage: a(2)
      0
      sage: a(3)
      1
      sage: a(11)
      360
      sage: a.list(12)
      [0, 0, 0, 1, 2, 5, 11, 23, 47, 94, 185, 360]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000100;a
      Number of compositions of n in which the maximum part size is 3.
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      0
      >>> a(Integer(2))
      0
      >>> a(Integer(3))
      1
      >>> a(Integer(11))
      360
      >>> a.list(Integer(12))
      [0, 0, 0, 1, 2, 5, 11, 23, 47, 94, 185, 360]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000108

  Bases: `SloaneSequence`

  Catalan numbers: \(C_n = \frac{\binom{2n}{n}}{n+1} = \frac{(2n)!}{n!(n+1)!}\).

  Also called Segner numbers.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000108;a
      Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.
      sage: a(0)
      1
      sage: a.offset
      0
      sage: a(8)
      1430
      sage: a(40)
      2622127042276492108820
      sage: a.list(9)
      [1, 1, 2, 5, 14, 42, 132, 429, 1430]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000108;a
      Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.
      >>> a(Integer(0))
      1
      >>> a.offset
      0
      >>> a(Integer(8))
      1430
      >>> a(Integer(40))
      2622127042276492108820
      >>> a.list(Integer(9))
      [1, 1, 2, 5, 14, 42, 132, 429, 1430]

  AUTHORS:

  - Jaap Spies (2007-01-12)

class sage.combinat.sloane_functions.A000110

  Bases: `ExponentialNumbers`

  The sequence of Bell numbers.

  The Bell number \(B_n\) counts the number of ways to put \(n\) distinguishable things into indistinguishable boxes such that no box is empty.

  Let \(S(n, k)\) denote the Stirling number of the second kind. Then

  \[B_n = \sum{k=0}^{n} S(n, k).\]

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; \(B_n\)

  EXAMPLES:

  Sage

      sage: a = sloane.A000110; a
      Sequence of Bell numbers
      sage: a.offset
      0
      sage: a(0)
      1
      sage: a(100)
      47585391276764833658790768841387207826363669686825611466616334637559114497892442622672724044217756306953557882560751
      sage: a.list(10)
      [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000110; a
      Sequence of Bell numbers
      >>> a.offset
      0
      >>> a(Integer(0))
      1
      >>> a(Integer(100))
      47585391276764833658790768841387207826363669686825611466616334637559114497892442622672724044217756306953557882560751
      >>> a.list(Integer(10))
      [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]

  AUTHORS:

  - Nick Alexander

class sage.combinat.sloane_functions.A000120

  Bases: `SloaneSequence`

  1s-counting sequence: number of 1s in binary expansion of \(n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000120;a
      1s-counting sequence: number of 1s in binary expansion of n.
      sage: a(0)
      0
      sage: a(2)
      1
      sage: a(12)
      2
      sage: a.list(12)
      [0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000120;a
      1s-counting sequence: number of 1s in binary expansion of n.
      >>> a(Integer(0))
      0
      >>> a(Integer(2))
      1
      >>> a(Integer(12))
      2
      >>> a.list(Integer(12))
      [0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3]

  AUTHORS:

  - Jaap Spies (2007-01-26)

  f(n)

    EXAMPLES:

    Sage

        sage: [sloane.A000120.f(n) for n in range(10)]
        [0, 1, 1, 2, 1, 2, 2, 3, 1, 2]

    Python

        >>> from sage.all import *
        >>> [sloane.A000120.f(n) for n in range(Integer(10))]
        [0, 1, 1, 2, 1, 2, 2, 3, 1, 2]

class sage.combinat.sloane_functions.A000124

  Bases: `SloaneSequence`

  Central polygonal numbers (the Lazy Caterer’s sequence): \(n(n+1)/2 + 1\).

  Or, maximal number of pieces formed when slicing a pancake with \(n\) cuts.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000124;a
      Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1.
      sage: a(0)
      1
      sage: a(1)
      2
      sage: a(2)
      4
      sage: a(9)
      46
      sage: a.list(10)
      [1, 2, 4, 7, 11, 16, 22, 29, 37, 46]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000124;a
      Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1.
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      2
      >>> a(Integer(2))
      4
      >>> a(Integer(9))
      46
      >>> a.list(Integer(10))
      [1, 2, 4, 7, 11, 16, 22, 29, 37, 46]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000129

  Bases: `RecurrenceSequence2`

  Pell numbers: \(a(0) = 0\), \(a(1) = 1\); for \(n > 1\), \(a(n) = 2a(n-1) + a(n-2)\).

  Denominators of continued fraction convergents to \(\sqrt 2\).

  See also A001333

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000129;a
      Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
      sage: a(0)
      0
      sage: a(2)
      2
      sage: a(12)
      13860
      sage: a.list(12)
      [0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000129;a
      Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
      >>> a(Integer(0))
      0
      >>> a(Integer(2))
      2
      >>> a(Integer(12))
      13860
      >>> a.list(Integer(12))
      [0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000142

  Bases: `SloaneSequence`

  Factorial numbers: \(n! = 1 \cdot 2 \cdot 3 \cdots n\).

  Order of symmetric group \(S_n\), number of permutations of \(n\) letters.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000142;a
      Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).
      sage: a(0)
      1
      sage: a(8)
      40320
      sage: a(40)
      815915283247897734345611269596115894272000000000
      sage: a.list(9)
      [1, 1, 2, 6, 24, 120, 720, 5040, 40320]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000142;a
      Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).
      >>> a(Integer(0))
      1
      >>> a(Integer(8))
      40320
      >>> a(Integer(40))
      815915283247897734345611269596115894272000000000
      >>> a.list(Integer(9))
      [1, 1, 2, 6, 24, 120, 720, 5040, 40320]

  AUTHORS:

  - Jaap Spies (2007-01-12)

class sage.combinat.sloane_functions.A000153

  Bases: `ExtremesOfPermanentsSequence`

  \(a(n) = n*a(n-1) + (n-2)*a(n-2)\), with \(a(0) = 0\), \(a(1) = 1\).

  With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=2\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000153; a
      a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(8)
      82508
      sage: a(20)
      10315043624498196944
      sage: a.list(8)
      [0, 1, 2, 7, 32, 181, 1214, 9403]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000153; a
      a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(8))
      82508
      >>> a(Integer(20))
      10315043624498196944
      >>> a.list(Integer(8))
      [0, 1, 2, 7, 32, 181, 1214, 9403]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A000165

  Bases: `SloaneSequence`

  Double factorial numbers: \((2n)!! = 2^n*n!\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000165;a
      Double factorial numbers: (2n)!! = 2^n*n!.
      sage: a(0)
      1
      sage: a.offset
      0
      sage: a(8)
      10321920
      sage: a(20)
      2551082656125828464640000
      sage: a.list(9)
      [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000165;a
      Double factorial numbers: (2n)!! = 2^n*n!.
      >>> a(Integer(0))
      1
      >>> a.offset
      0
      >>> a(Integer(8))
      10321920
      >>> a(Integer(20))
      2551082656125828464640000
      >>> a.list(Integer(9))
      [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920]

  AUTHORS:

  - Jaap Spies (2007-01-24)

class sage.combinat.sloane_functions.A000166

  Bases: `SloaneSequence`

  Subfactorial or rencontres numbers, or derangements: number of permutations of \(n\) elements with no fixed points.

  With offset 1 also the permanent of a (0,1)-matrix of order \(n\) with \(n\) 0s not on a line.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000166;a
      Subfactorial or rencontres numbers, or derangements: number of permutations of $n$ elements with no fixed points.
      sage: a(0)
      1
      sage: a(1)
      0
      sage: a(2)
      1
      sage: a.offset
      0
      sage: a(8)
      14833
      sage: a(20)
      895014631192902121
      sage: a.list(9)
      [1, 0, 1, 2, 9, 44, 265, 1854, 14833]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000166;a
      Subfactorial or rencontres numbers, or derangements: number of permutations of $n$ elements with no fixed points.
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      0
      >>> a(Integer(2))
      1
      >>> a.offset
      0
      >>> a(Integer(8))
      14833
      >>> a(Integer(20))
      895014631192902121
      >>> a.list(Integer(9))
      [1, 0, 1, 2, 9, 44, 265, 1854, 14833]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A000169

  Bases: `SloaneSequence`

  Number of labeled rooted trees with \(n\) nodes: \(n^{(n-1)}\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000169;a
      Number of labeled rooted trees with n nodes: n^(n-1).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(10)
      1000000000
      sage: a.list(11)
      [1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000169;a
      Number of labeled rooted trees with n nodes: n^(n-1).
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      2
      >>> a(Integer(10))
      1000000000
      >>> a.list(Integer(11))
      [1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000203

  Bases: `SloaneSequence`

  The sequence \(\sigma(n)\), where \(\sigma(n)\) is the sum of the divisors of \(n\). Also called \(\sigma_1(n)\).

  The function `sigma(n, k)` implements \(\sigma_k(n)\) in Sage.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000203; a
      sigma(n) = sum of divisors of n. Also called sigma_1(n).
      sage: a(1)
      1
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(256)
      511
      sage: a.list(12)
      [1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28]
      sage: a(1/3)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  Python

      >>> from sage.all import *
      >>> a = sloane.A000203; a
      sigma(n) = sum of divisors of n. Also called sigma_1(n).
      >>> a(Integer(1))
      1
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(256))
      511
      >>> a.list(Integer(12))
      [1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28]
      >>> a(Integer(1)/Integer(3))
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  AUTHORS:

  - Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A000204

  Bases: `SloaneSequence`

  Lucas numbers (beginning with 1): \(L(n) = L(n-1) + L(n-2)\) with \(L(1) = 1\), \(L(2) = 3\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000204; a
      Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3.
      sage: a(1)
      1
      sage: a(8)
      47
      sage: a(200)
      627376215338105766356982006981782561278127
      sage: a(-4)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be a positive integer
      sage: a.list(12)
      [1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322]
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  Python

      >>> from sage.all import *
      >>> a = sloane.A000204; a
      Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3.
      >>> a(Integer(1))
      1
      >>> a(Integer(8))
      47
      >>> a(Integer(200))
      627376215338105766356982006981782561278127
      >>> a(-Integer(4))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be a positive integer
      >>> a.list(Integer(12))
      [1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322]
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  AUTHORS:

  - Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000213

  Bases: `SloaneSequence`

  Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 1, 1, 1, …

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000213;a
      Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(2)
      1
      sage: a(11)
      355
      sage: a.list(12)
      [1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000213;a
      Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      1
      >>> a(Integer(11))
      355
      >>> a.list(Integer(12))
      [1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355]

  AUTHORS:

  - Jaap Spies (2007-01-19)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A000213.list(10)
        [1, 1, 1, 3, 5, 9, 17, 31, 57, 105]

    Python

        >>> from sage.all import *
        >>> sloane.A000213.list(Integer(10))
        [1, 1, 1, 3, 5, 9, 17, 31, 57, 105]

class sage.combinat.sloane_functions.A000217

  Bases: `SloaneSequence`

  Triangular numbers: \(a(n) = \binom{n+1}{2} = n(n+1)/2\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000217;a
      Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
      sage: a(0)
      0
      sage: a(2)
      3
      sage: a(8)
      36
      sage: a(2000)
      2001000
      sage: a.list(9)
      [0, 1, 3, 6, 10, 15, 21, 28, 36]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000217;a
      Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
      >>> a(Integer(0))
      0
      >>> a(Integer(2))
      3
      >>> a(Integer(8))
      36
      >>> a(Integer(2000))
      2001000
      >>> a.list(Integer(9))
      [0, 1, 3, 6, 10, 15, 21, 28, 36]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000225

  Bases: `SloaneSequence`

  \(2^n - 1\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000225;a
      2^n - 1.
      sage: a(0)
      0
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(12)
      4095
      sage: a.list(12)
      [0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000225;a
      2^n - 1.
      >>> a(Integer(0))
      0
      >>> a(-Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      >>> a(Integer(12))
      4095
      >>> a.list(Integer(12))
      [0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000244

  Bases: `SloaneSequence`

  Powers of 3: \(a(n) = 3^n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000244;a
      Powers of 3: a(n) = 3^n.
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      1
      sage: a(3)
      27
      sage: a(11)
      177147
      sage: a.list(12)
      [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000244;a
      Powers of 3: a(n) = 3^n.
      >>> a(-Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      >>> a(Integer(0))
      1
      >>> a(Integer(3))
      27
      >>> a(Integer(11))
      177147
      >>> a.list(Integer(12))
      [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000255

  Bases: `ExtremesOfPermanentsSequence`

  \(a(n) = n*a(n-1) + (n-1)*a(n-2)\), with \(a(0) = 1\), \(a(1) = 1\).

  With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=1\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000255;a
      a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a.offset
      0
      sage: a(8)
      148329
      sage: a(22)
      9923922230666898717143
      sage: a.list(9)
      [1, 1, 3, 11, 53, 309, 2119, 16687, 148329]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000255;a
      a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      1
      >>> a.offset
      0
      >>> a(Integer(8))
      148329
      >>> a(Integer(22))
      9923922230666898717143
      >>> a.list(Integer(9))
      [1, 1, 3, 11, 53, 309, 2119, 16687, 148329]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A000261

  Bases: `ExtremesOfPermanentsSequence`

  \(a(n) = n*a(n-1) + (n-3)*a(n-2)\), with \(a(1) = 1\), \(a(2) = 1\).

  With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=3\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000261;a
      a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      0
      sage: a.offset
      1
      sage: a(8)
      30637
      sage: a(22)
      1801366114380914335441
      sage: a.list(9)
      [0, 1, 3, 13, 71, 465, 3539, 30637, 296967]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000261;a
      a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      0
      >>> a.offset
      1
      >>> a(Integer(8))
      30637
      >>> a(Integer(22))
      1801366114380914335441
      >>> a.list(Integer(9))
      [0, 1, 3, 13, 71, 465, 3539, 30637, 296967]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A000272

  Bases: `SloaneSequence`

  Number of labeled rooted trees on \(n\) nodes: \(n^{(n-2)}\).

  INPUT:

  - `n` – integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000272;a
      Number of labeled rooted trees with n nodes: n^(n-2).
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(2)
      1
      sage: a(10)
      100000000
      sage: a.list(12)
      [1, 1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000272;a
      Number of labeled rooted trees with n nodes: n^(n-2).
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      1
      >>> a(Integer(10))
      100000000
      >>> a.list(Integer(12))
      [1, 1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000290

  Bases: `SloaneSequence`

  The squares: \(a(n) = n^2\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000290;a
      The squares: a(n) = n^2.
      sage: a(0)
      0
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(16)
      256
      sage: a.list(17)
      [0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000290;a
      The squares: a(n) = n^2.
      >>> a(Integer(0))
      0
      >>> a(-Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      >>> a(Integer(16))
      256
      >>> a.list(Integer(17))
      [0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000292

  Bases: `SloaneSequence`

  Tetrahedral (or pyramidal) numbers: \(\binom{n+2}{3} = n(n+1)(n+2)/6\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000292;a
      Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.
      sage: a(0)
      0
      sage: a(2)
      4
      sage: a(11)
      286
      sage: a.list(12)
      [0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000292;a
      Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.
      >>> a(Integer(0))
      0
      >>> a(Integer(2))
      4
      >>> a(Integer(11))
      286
      >>> a.list(Integer(12))
      [0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000302

  Bases: `SloaneSequence`

  Powers of 4: \(a(n) = 4^n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000302;a
      Powers of 4: a(n) = 4^n.
      sage: a(0)
      1
      sage: a(1)
      4
      sage: a(2)
      16
      sage: a(10)
      1048576
      sage: a.list(12)
      [1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000302;a
      Powers of 4: a(n) = 4^n.
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      4
      >>> a(Integer(2))
      16
      >>> a(Integer(10))
      1048576
      >>> a.list(Integer(12))
      [1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000312

  Bases: `SloaneSequence`

  Number of labeled mappings from \(n\) points to themselves (endofunctions): \(n^n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000312;a
      Number of labeled mappings from n points to themselves (endofunctions): n^n.
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(9)
      387420489
      sage: a.list(11)
      [1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000312;a
      Number of labeled mappings from n points to themselves (endofunctions): n^n.
      >>> a(-Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      1
      >>> a(Integer(9))
      387420489
      >>> a.list(Integer(11))
      [1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000326

  Bases: `SloaneSequence`

  Pentagonal numbers: \(n(3n-1)/2\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000326;a
      Pentagonal numbers: n(3n-1)/2.
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      5
      sage: a(10)
      145
      sage: a.list(12)
      [0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176]
      sage: a(1/3)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  Python

      >>> from sage.all import *
      >>> a = sloane.A000326;a
      Pentagonal numbers: n(3n-1)/2.
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      5
      >>> a(Integer(10))
      145
      >>> a.list(Integer(12))
      [0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176]
      >>> a(Integer(1)/Integer(3))
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000330

  Bases: `SloaneSequence`

  Square pyramidal numbers” \(0^2 + 1^2 \cdots n^2 = n(n+1)(2n+1)/6\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000330;a
      Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      0
      sage: a(3)
      14
      sage: a(11)
      506
      sage: a.list(12)
      [0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000330;a
      Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.
      >>> a(-Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      >>> a(Integer(0))
      0
      >>> a(Integer(3))
      14
      >>> a(Integer(11))
      506
      >>> a.list(Integer(12))
      [0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000396

  Bases: `SloaneSequence`

  Perfect numbers: equal to sum of proper divisors.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000396;a
      Perfect numbers: equal to sum of proper divisors.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      6
      sage: a(2)
      28
      sage: a(7)
      137438691328
      sage: a.list(7)
      [6, 28, 496, 8128, 33550336, 8589869056, 137438691328]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000396;a
      Perfect numbers: equal to sum of proper divisors.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      6
      >>> a(Integer(2))
      28
      >>> a(Integer(7))
      137438691328
      >>> a.list(Integer(7))
      [6, 28, 496, 8128, 33550336, 8589869056, 137438691328]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000578

  Bases: `SloaneSequence`

  The cubes: \(a(n) = n^3\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000578;a
      The cubes: n^3
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      0
      sage: a(3)
      27
      sage: a(11)
      1331
      sage: a.list(12)
      [0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000578;a
      The cubes: n^3
      >>> a(-Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      >>> a(Integer(0))
      0
      >>> a(Integer(3))
      27
      >>> a(Integer(11))
      1331
      >>> a.list(Integer(12))
      [0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000583

  Bases: `SloaneSequence`

  Fourth powers: \(a(n) = n^4\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000583;a
      Fourth powers: n^4.
      sage: a(0.0)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer
      sage: a(1)
      1
      sage: a(2)
      16
      sage: a(9)
      6561
      sage: a.list(10)
      [0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000583;a
      Fourth powers: n^4.
      >>> a(RealNumber('0.0'))
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      16
      >>> a(Integer(9))
      6561
      >>> a.list(Integer(10))
      [0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561]

  AUTHORS:

  - Jaap Spies (2007-02-04)

class sage.combinat.sloane_functions.A000587

  Bases: `ExponentialNumbers`

  The sequence of Uppuluri-Carpenter numbers.

  The Uppuluri-Carpenter number \(C_n\) counts the imbalance in the number of ways to put \(n\) distinguishable things into an even number of indistinguishable boxes versus into an odd number of indistinguishable boxes, such that no box is empty.

  Let \(S(n, k)\) denote the Stirling number of the second kind. Then

  \[C_n = \sum{k=0}^{n} (-1)^k S(n, k) .\]

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; \(C_n\)

  EXAMPLES:

  Sage

      sage: a = sloane.A000587; a
      Sequence of Uppuluri-Carpenter numbers
      sage: a.offset
      0
      sage: a(0)
      1
      sage: a(100)
      397577026456518507969762382254187048845620355238545130875069912944235105204434466095862371032124545552161
      sage: a.list(10)
      [1, -1, 0, 1, 1, -2, -9, -9, 50, 267]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000587; a
      Sequence of Uppuluri-Carpenter numbers
      >>> a.offset
      0
      >>> a(Integer(0))
      1
      >>> a(Integer(100))
      397577026456518507969762382254187048845620355238545130875069912944235105204434466095862371032124545552161
      >>> a.list(Integer(10))
      [1, -1, 0, 1, 1, -2, -9, -9, 50, 267]

  AUTHORS:

  - Nick Alexander

class sage.combinat.sloane_functions.A000668

  Bases: `SloaneSequence`

  Mersenne primes (of form \(2^p - 1\) where \(p\) is a prime).

  (See A000043 for the values of \(p\).)

  Warning: a(39) has 4,053,946 digits!

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000668;a
      Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the values of p.)
      sage: a(1)
      3
      sage: a(2)
      7
      sage: a(12)
      170141183460469231731687303715884105727

  Python

      >>> from sage.all import *
      >>> a = sloane.A000668;a
      Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the values of p.)
      >>> a(Integer(1))
      3
      >>> a(Integer(2))
      7
      >>> a(Integer(12))
      170141183460469231731687303715884105727

  Warning: a(39) has 4,053,946 digits!

  Sage

      sage: a(40)
      Traceback (most recent call last):
      ...
      IndexError: list index out of range
      sage: a.list(8)
      [3, 7, 31, 127, 8191, 131071, 524287, 2147483647]

  [Python]

      >>> from sage.all import *
      >>> a(Integer(40))
      Traceback (most recent call last):
      ...
      IndexError: list index out of range
      >>> a.list(Integer(8))
      [3, 7, 31, 127, 8191, 131071, 524287, 2147483647]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000670

  Bases: `SloaneSequence`

  Number of preferential arrangements of \(n\) labeled elements; or number of weak orders on \(n\) labeled elements.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000670;a
      Number of preferential arrangements of n labeled elements.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(2)
      3
      sage: a(9)
      7087261
      sage: a.list(10)
      [1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000670;a
      Number of preferential arrangements of n labeled elements.
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      3
      >>> a(Integer(9))
      7087261
      >>> a.list(Integer(10))
      [1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]

  AUTHORS:

  - Jaap Spies (2007-02-03)

class sage.combinat.sloane_functions.A000720

  Bases: `SloaneSequence`

  \(pi(n)\), the number of primes \(\le n\). Sometimes called \(PrimePi(n)\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000720;a
      pi(n), the number of primes <= n. Sometimes called PrimePi(n)
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      1
      sage: a(8)
      4
      sage: a(1000)
      168
      sage: a.list(12)
      [0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000720;a
      pi(n), the number of primes <= n. Sometimes called PrimePi(n)
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(2))
      1
      >>> a(Integer(8))
      4
      >>> a(Integer(1000))
      168
      >>> a.list(Integer(12))
      [0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000796

  Bases: `SloaneSequence`

  Decimal expansion of \(\pi\).

  INPUT:

  - `n` – positive integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000796;a
      Decimal expansion of Pi.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      3
      sage: a(13)
      9
      sage: a.list(14)
      [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7]
      sage: a(100)
      7

  Python

      >>> from sage.all import *
      >>> a = sloane.A000796;a
      Decimal expansion of Pi.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      3
      >>> a(Integer(13))
      9
      >>> a.list(Integer(14))
      [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7]
      >>> a(Integer(100))
      7

  AUTHOR:

  - Jaap Spies (2007-01-30)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A000796.list(10)
        [3, 1, 4, 1, 5, 9, 2, 6, 5, 3]

    Python

        >>> from sage.all import *
        >>> sloane.A000796.list(Integer(10))
        [3, 1, 4, 1, 5, 9, 2, 6, 5, 3]

  pi()

    Based on an algorithm of Lambert Meertens The ABC-programming language!!!

    EXAMPLES:

    Sage

        sage: it = sloane.A000796.pi()
        sage: [next(it) for i in range(10)]
        [3, 1, 4, 1, 5, 9, 2, 6, 5, 3]

    Python

        >>> from sage.all import *
        >>> it = sloane.A000796.pi()
        >>> [next(it) for i in range(Integer(10))]
        [3, 1, 4, 1, 5, 9, 2, 6, 5, 3]

class sage.combinat.sloane_functions.A000961

  Bases: `SloaneSequence`

  Prime powers.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000961;a
      Prime powers.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      2
      sage: a(12)
      17
      sage: a.list(12)
      [1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000961;a
      Prime powers.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(2))
      2
      >>> a(Integer(12))
      17
      >>> a.list(Integer(12))
      [1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17]

  AUTHORS:

  - Jaap Spies (2007-01-25)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A000961.list(10)
        [1, 2, 3, 4, 5, 7, 8, 9, 11, 13]

    Python

        >>> from sage.all import *
        >>> sloane.A000961.list(Integer(10))
        [1, 2, 3, 4, 5, 7, 8, 9, 11, 13]

class sage.combinat.sloane_functions.A000984

  Bases: `SloaneSequence`

  Central binomial coefficients: \(\binom{2n}{n} = \frac {(2n)!} {(n!)^2}\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A000984;a
      Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2
      sage: a(0)
      1
      sage: a(2)
      6
      sage: a(8)
      12870
      sage: a.list(9)
      [1, 2, 6, 20, 70, 252, 924, 3432, 12870]

  Python

      >>> from sage.all import *
      >>> a = sloane.A000984;a
      Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2
      >>> a(Integer(0))
      1
      >>> a(Integer(2))
      6
      >>> a(Integer(8))
      12870
      >>> a.list(Integer(9))
      [1, 2, 6, 20, 70, 252, 924, 3432, 12870]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A001006

  Bases: `SloaneSequence`

  Motzkin numbers: number of ways of drawing any number of nonintersecting chords among \(n\) points on a circle.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001006;a
      Motzkin numbers: number of ways of drawing any number of nonintersecting chords among n points on a circle.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(12)
      15511
      sage: a.list(13)
      [1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001006;a
      Motzkin numbers: number of ways of drawing any number of nonintersecting chords among n points on a circle.
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      2
      >>> a(Integer(12))
      15511
      >>> a.list(Integer(13))
      [1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511]

  AUTHORS:

  - Jaap Spies (2007-02-02)

class sage.combinat.sloane_functions.A001045

  Bases: `RecurrenceSequence2`

  Jacobsthal sequence: \(a(n) = a(n-1) + 2a(n-2)\), \(a(0) = 0\) and \(a(1) = 1\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001045;a
      Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      1
      sage: a(11)
      683
      sage: a.list(12)
      [0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001045;a
      Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      1
      >>> a(Integer(11))
      683
      >>> a.list(Integer(12))
      [0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A001055

  Bases: `SloaneSequence`

  Number of ways of factoring \(n\) with all factors 1.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001055;a
      Number of ways of factoring n with all factors >1.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      1
      sage: a(9)
      2
      sage: a.list(16)
      [1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001055;a
      Number of ways of factoring n with all factors >1.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      1
      >>> a(Integer(9))
      2
      >>> a.list(Integer(16))
      [1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5]

  AUTHORS:

  - Jaap Spies (2007-02-04)

  nwf(n, m)

    EXAMPLES:

    Sage

        sage: sloane.A001055.nwf(4,1)
        0
        sage: sloane.A001055.nwf(4,2)
        1
        sage: sloane.A001055.nwf(4,3)
        1
        sage: sloane.A001055.nwf(4,4)
        2

    Python

        >>> from sage.all import *
        >>> sloane.A001055.nwf(Integer(4),Integer(1))
        0
        >>> sloane.A001055.nwf(Integer(4),Integer(2))
        1
        >>> sloane.A001055.nwf(Integer(4),Integer(3))
        1
        >>> sloane.A001055.nwf(Integer(4),Integer(4))
        2

class sage.combinat.sloane_functions.A001109

  Bases: `RecurrenceSequence2`

  \(a(n)^2\) is a triangular number: \(a(n) = 6*a(n-1) - a(n-2)\) with \(a(0)=0\), \(a(1)=1\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001109;a
      a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      6
      sage: a.offset
      0
      sage: a(8)
      235416
      sage: a(60)
      1515330104844857898115857393785728383101709300
      sage: a.list(9)
      [0, 1, 6, 35, 204, 1189, 6930, 40391, 235416]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001109;a
      a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      6
      >>> a.offset
      0
      >>> a(Integer(8))
      235416
      >>> a(Integer(60))
      1515330104844857898115857393785728383101709300
      >>> a.list(Integer(9))
      [0, 1, 6, 35, 204, 1189, 6930, 40391, 235416]

  AUTHORS:

  - Jaap Spies (2007-01-24)

class sage.combinat.sloane_functions.A001110

  Bases: `RecurrenceSequence`

  Numbers that are both triangular and square: \(a(n) = 34a(n-1) - a(n-2) + 2\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001110; a
      Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2.
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(8)
      55420693056
      sage: a(21)
      4446390382511295358038307980025
      sage: a.list(8)
      [0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001110; a
      Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2.
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(8))
      55420693056
      >>> a(Integer(21))
      4446390382511295358038307980025
      >>> a.list(Integer(8))
      [0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]

  AUTHORS:

  - Jaap Spies (2007-01-19)

  g(k)

    EXAMPLES:

    Sage

        sage: sloane.A001110.g(2)
        2
        sage: sloane.A001110.g(1)
        0

    Python

        >>> from sage.all import *
        >>> sloane.A001110.g(Integer(2))
        2
        >>> sloane.A001110.g(Integer(1))
        0

  link = 'http://oeis.org/classic/A001110'

class sage.combinat.sloane_functions.A001147

  Bases: `SloaneSequence`

  Double factorial numbers: \((2n-1)!! = 1 \cdot 3 \cdot 5 \cdots (2n-1)\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001147;a
      Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).
      sage: a(0)
      1
      sage: a.offset
      0
      sage: a(8)
      2027025
      sage: a(20)
      319830986772877770815625
      sage: a.list(9)
      [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001147;a
      Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).
      >>> a(Integer(0))
      1
      >>> a.offset
      0
      >>> a(Integer(8))
      2027025
      >>> a(Integer(20))
      319830986772877770815625
      >>> a.list(Integer(9))
      [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025]

  AUTHORS:

  - Jaap Spies (2007-01-24)

class sage.combinat.sloane_functions.A001157

  Bases: `SloaneSequence`

  The sequence \(\sigma_2(n)\), sum of squares of divisors of \(n\).

  The function sigma(n, k) implements \(\sigma_k*\) in Sage.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001157;a
      sigma_2(n): sum of squares of divisors of n
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      5
      sage: a(8)
      85
      sage: a.list(9)
      [1, 5, 10, 21, 26, 50, 50, 85, 91]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001157;a
      sigma_2(n): sum of squares of divisors of n
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(2))
      5
      >>> a(Integer(8))
      85
      >>> a.list(Integer(9))
      [1, 5, 10, 21, 26, 50, 50, 85, 91]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A001189

  Bases: `SloaneSequence`

  Number of degree-n permutations of order exactly 2.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001189;a
      Number of degree-n permutations of order exactly 2.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      0
      sage: a(2)
      1
      sage: a(12)
      140151
      sage: a.list(13)
      [0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001189;a
      Number of degree-n permutations of order exactly 2.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      0
      >>> a(Integer(2))
      1
      >>> a(Integer(12))
      140151
      >>> a.list(Integer(13))
      [0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503]

  AUTHORS:

  - Jaap Spies (2007-02-03)

class sage.combinat.sloane_functions.A001221

  Bases: `SloaneSequence`

  Number of different prime divisors of \(n\).

  Also called omega(n) or \(\omega(n)\). Maximal number of terms in any factorization of \(n\). Number of prime powers that divide \(n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001221; a
      Number of distinct primes dividing n (also called omega(n)).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      0
      sage: a(8)
      1
      sage: a(41)
      1
      sage: a(84792)
      3
      sage: a.list(12)
      [0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001221; a
      Number of distinct primes dividing n (also called omega(n)).
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      0
      >>> a(Integer(8))
      1
      >>> a(Integer(41))
      1
      >>> a(Integer(84792))
      3
      >>> a.list(Integer(12))
      [0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A001222

  Bases: `SloaneSequence`

  Number of prime divisors of \(n\) (counted with multiplicity).

  Also called bigomega(n) or \(\Omega(n)\). Maximal number of terms in any factorization of \(n\). Number of prime powers that divide \(n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001222; a
      Number of prime divisors of n (counted with multiplicity).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      0
      sage: a(8)
      3
      sage: a(41)
      1
      sage: a(84792)
      5
      sage: a.list(12)
      [0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001222; a
      Number of prime divisors of n (counted with multiplicity).
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      0
      >>> a(Integer(8))
      3
      >>> a(Integer(41))
      1
      >>> a(Integer(84792))
      5
      >>> a.list(Integer(12))
      [0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A001227

  Bases: `SloaneSequence`

  Number of odd divisors of \(n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001227; a
      Number of odd divisors of n
      sage: a.offset
      1
      sage: a(1)
      1
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(100)
      3
      sage: a(256)
      1
      sage: a(29)
      2
      sage: a.list(20)
      [1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2]
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be a positive integer

  Python

      >>> from sage.all import *
      >>> a = sloane.A001227; a
      Number of odd divisors of n
      >>> a.offset
      1
      >>> a(Integer(1))
      1
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(100))
      3
      >>> a(Integer(256))
      1
      >>> a(Integer(29))
      2
      >>> a.list(Integer(20))
      [1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2]
      >>> a(-Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be a positive integer

  AUTHORS:

  - Jaap Spies (2007-01-14)

class sage.combinat.sloane_functions.A001333

  Bases: `RecurrenceSequence2`

  Numerators of continued fraction convergents to \(\sqrt 2\).

  See also A000129

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001333;a
      Numerators of continued fraction convergents to sqrt(2).
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(2)
      3
      sage: a(3)
      7
      sage: a(11)
      8119
      sage: a.list(12)
      [1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001333;a
      Numerators of continued fraction convergents to sqrt(2).
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      3
      >>> a(Integer(3))
      7
      >>> a(Integer(11))
      8119
      >>> a.list(Integer(12))
      [1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119]

  AUTHORS:

  - Jaap Spies (2007-02-01)

class sage.combinat.sloane_functions.A001358

  Bases: `SloaneSequence`

  Products of two primes.

  These numbers have been called semiprimes (or semi-primes), biprimes or 2-almost primes.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001358;a
      Products of two primes.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      6
      sage: a(8)
      22
      sage: a(200)
      669
      sage: a.list(9)
      [4, 6, 9, 10, 14, 15, 21, 22, 25]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001358;a
      Products of two primes.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(2))
      6
      >>> a(Integer(8))
      22
      >>> a(Integer(200))
      669
      >>> a.list(Integer(9))
      [4, 6, 9, 10, 14, 15, 21, 22, 25]

  AUTHORS:

  - Jaap Spies (2007-01-25)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A001358.list(9)
        [4, 6, 9, 10, 14, 15, 21, 22, 25]

    Python

        >>> from sage.all import *
        >>> sloane.A001358.list(Integer(9))
        [4, 6, 9, 10, 14, 15, 21, 22, 25]

class sage.combinat.sloane_functions.A001405

  Bases: `SloaneSequence`

  Central binomial coefficients: \(\binom{n}{\lfloor \frac {n}{ 2} \rfloor}\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001405;a
      Central binomial coefficients: C(n,floor(n/2)).
      sage: a(0)
      1
      sage: a(2)
      2
      sage: a(12)
      924
      sage: a.list(12)
      [1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001405;a
      Central binomial coefficients: C(n,floor(n/2)).
      >>> a(Integer(0))
      1
      >>> a(Integer(2))
      2
      >>> a(Integer(12))
      924
      >>> a.list(Integer(12))
      [1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A001477

  Bases: `SloaneSequence`

  The nonnegative integers.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001477;a
      The nonnegative integers.
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      0
      sage: a(3382789)
      3382789
      sage: a(11)
      11
      sage: a.list(12)
      [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001477;a
      The nonnegative integers.
      >>> a(-Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      >>> a(Integer(0))
      0
      >>> a(Integer(3382789))
      3382789
      >>> a(Integer(11))
      11
      >>> a.list(Integer(12))
      [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A001694

  Bases: `SloaneSequence`

  This function returns the \(n\)-th Powerful Number:

  A positive integer \(n\) is powerful if for every prime \(p\) dividing \(n\), \(p^2\) also divides \(n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001694; a
      Powerful Numbers (also called squarefull, square-full or 2-full numbers).
      sage: a.offset
      1
      sage: a(1)
      1
      sage: a(4)
      9
      sage: a(100)
      3136
      sage: a(156)
      7225
      sage: a.list(19)
      [1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144]
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be a positive integer

  Python

      >>> from sage.all import *
      >>> a = sloane.A001694; a
      Powerful Numbers (also called squarefull, square-full or 2-full numbers).
      >>> a.offset
      1
      >>> a(Integer(1))
      1
      >>> a(Integer(4))
      9
      >>> a(Integer(100))
      3136
      >>> a(Integer(156))
      7225
      >>> a.list(Integer(19))
      [1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144]
      >>> a(-Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be a positive integer

  AUTHORS:

  - Jaap Spies (2007-01-14)

  is_powerful(n)

    Return `True` if and only if \(n\) is a powerful number.

    A positive integer \(n\) is powerful if for every prime \(p\) dividing \(n\), \(p^2\) also divides \(n\).

    See OEIS sequence A001694.

    INPUT:

    - `n` – integer

    OUTPUT: `True` if \(n\) is a powerful number, else `False`

    EXAMPLES:

    Sage

        sage: a = sloane.A001694
        sage: a.is_powerful(2500)
        True
        sage: a.is_powerful(20)
        False

    Python

        >>> from sage.all import *
        >>> a = sloane.A001694
        >>> a.is_powerful(Integer(2500))
        True
        >>> a.is_powerful(Integer(20))
        False

    AUTHORS:

    - Jaap Spies (2006-12-07)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A001694.list(9)
        [1, 4, 8, 9, 16, 25, 27, 32, 36]

    Python

        >>> from sage.all import *
        >>> sloane.A001694.list(Integer(9))
        [1, 4, 8, 9, 16, 25, 27, 32, 36]

class sage.combinat.sloane_functions.A001836

  Bases: `SloaneSequence`

  Numbers \(n\) such that \(\phi(2n-1) < \phi(2n)\), where \(\phi\) is Euler’s totient function.

  Euler’s totient function is also known as `euler_phi`; `euler_phi` is a standard Sage function.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001836; a
      Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient function A000010.
      sage: a.offset
      1
      sage: a(1)
      53
      sage: a(8)
      683
      sage: a(300)
      17798
      sage: a.list(12)
      [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  Python

      >>> from sage.all import *
      >>> a = sloane.A001836; a
      Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient function A000010.
      >>> a.offset
      1
      >>> a(Integer(1))
      53
      >>> a(Integer(8))
      683
      >>> a(Integer(300))
      17798
      >>> a.list(Integer(12))
      [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  Compare: Searching Sloane’s online database… Numbers n such that phi(2n-1) phi(2n), where phi is Euler’s totient function A000010. [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]

  AUTHORS:

  - Jaap Spies (2007-01-17)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A001836.list(9)
        [53, 83, 158, 263, 293, 368, 578, 683, 743]

    Python

        >>> from sage.all import *
        >>> sloane.A001836.list(Integer(9))
        [53, 83, 158, 263, 293, 368, 578, 683, 743]

class sage.combinat.sloane_functions.A001906

  Bases: `RecurrenceSequence2`

  \(F(2n) =\) bisection of Fibonacci sequence: \(a(n)=3a(n-1)-a(n-2)\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001906; a
      F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(8)
      987
      sage: a(22)
      701408733
      sage: a.list(12)
      [0, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001906; a
      F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(8))
      987
      >>> a(Integer(22))
      701408733
      >>> a.list(Integer(12))
      [0, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A001909

  Bases: `ExtremesOfPermanentsSequence`

  \(a(n) = n*a(n-1) + (n-4)*a(n-2)\), with \(a(2) = 0\), \(a(3) = 1\).

  With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=4\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` – positive integer >= 2

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001909;a
      a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1.
      sage: a(1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=1) must be an integer >= 2
      sage: a.offset
      2
      sage: a(2)
      0
      sage: a(8)
      8544
      sage: a(22)
      470033715095287415734
      sage: a.list(9)
      [0, 1, 4, 21, 134, 1001, 8544, 81901, 870274]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001909;a
      a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1.
      >>> a(Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=1) must be an integer >= 2
      >>> a.offset
      2
      >>> a(Integer(2))
      0
      >>> a(Integer(8))
      8544
      >>> a(Integer(22))
      470033715095287415734
      >>> a.list(Integer(9))
      [0, 1, 4, 21, 134, 1001, 8544, 81901, 870274]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A001910

  Bases: `ExtremesOfPermanentsSequence`

  \(a(n) = n*a(n-1) + (n-5)*a(n-2)\), with \(a(3) = 0\), \(a(4) = 1\).

  With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=5\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` – positive integer \(\geq 3\)

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001910;a
      a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be an integer >= 3
      sage: a(3)
      0
      sage: a.offset
      3
      sage: a(8)
      1909
      sage: a(22)
      98125321641110663023
      sage: a.list(9)
      [0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001910;a
      a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be an integer >= 3
      >>> a(Integer(3))
      0
      >>> a.offset
      3
      >>> a(Integer(8))
      1909
      >>> a(Integer(22))
      98125321641110663023
      >>> a.list(Integer(9))
      [0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A001969

  Bases: `SloaneSequence`

  Evil numbers: even number of 1s in binary expansion.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A001969;a
      Evil numbers: even number of 1s in binary expansion.
      sage: a(0)
      0
      sage: a(1)
      3
      sage: a(2)
      5
      sage: a(12)
      24
      sage: a.list(13)
      [0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24]

  Python

      >>> from sage.all import *
      >>> a = sloane.A001969;a
      Evil numbers: even number of 1s in binary expansion.
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      3
      >>> a(Integer(2))
      5
      >>> a(Integer(12))
      24
      >>> a.list(Integer(13))
      [0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24]

  AUTHORS:

  - Jaap Spies (2007-02-02)

class sage.combinat.sloane_functions.A002110

  Bases: `SloaneSequence`

  Primorial numbers (first definition): product of first \(n\) primes. Sometimes written \(p\#\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A002110;a
      Primorial numbers (first definition): product of first n primes. Sometimes written p#.
      sage: a(0)
      1
      sage: a(2)
      6
      sage: a(8)
      9699690
      sage: a(17)
      1922760350154212639070
      sage: a.list(9)
      [1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690]

  Python

      >>> from sage.all import *
      >>> a = sloane.A002110;a
      Primorial numbers (first definition): product of first n primes. Sometimes written p#.
      >>> a(Integer(0))
      1
      >>> a(Integer(2))
      6
      >>> a(Integer(8))
      9699690
      >>> a(Integer(17))
      1922760350154212639070
      >>> a.list(Integer(9))
      [1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A002113

  Bases: `SloaneSequence`

  Palindromes in base 10.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A002113;a
      Palindromes in base 10.
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(12)
      33
      sage: a.list(13)
      [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33]

  Python

      >>> from sage.all import *
      >>> a = sloane.A002113;a
      Palindromes in base 10.
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      2
      >>> a(Integer(12))
      33
      >>> a.list(Integer(13))
      [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33]

  AUTHORS:

  - Jaap Spies (2007-02-02)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A002113.list(15)
        [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55]

    Python

        >>> from sage.all import *
        >>> sloane.A002113.list(Integer(15))
        [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55]

class sage.combinat.sloane_functions.A002275

  Bases: `SloaneSequence`

  Repunits: \(\frac {(10^n - 1)}{9}\). Often denoted by \(R_n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A002275;a
      Repunits: (10^n - 1)/9. Often denoted by R_n.
      sage: a(0)
      0
      sage: a(2)
      11
      sage: a(8)
      11111111
      sage: a(20)
      11111111111111111111
      sage: a.list(9)
      [0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111]

  Python

      >>> from sage.all import *
      >>> a = sloane.A002275;a
      Repunits: (10^n - 1)/9. Often denoted by R_n.
      >>> a(Integer(0))
      0
      >>> a(Integer(2))
      11
      >>> a(Integer(8))
      11111111
      >>> a(Integer(20))
      11111111111111111111
      >>> a.list(Integer(9))
      [0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A002378

  Bases: `SloaneSequence`

  Oblong (or pronic, or heteromecic) numbers: \(n(n+1)\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A002378;a
      Oblong (or pronic, or heteromecic) numbers: n(n+1).
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      0
      sage: a(1)
      2
      sage: a(11)
      132
      sage: a.list(12)
      [0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132]

  Python

      >>> from sage.all import *
      >>> a = sloane.A002378;a
      Oblong (or pronic, or heteromecic) numbers: n(n+1).
      >>> a(-Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      2
      >>> a(Integer(11))
      132
      >>> a.list(Integer(12))
      [0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A002620

  Bases: `SloaneSequence`

  Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, \(\lfloor n^2/4 \rfloor\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A002620;a
      Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).
      sage: a(0)
      0
      sage: a(1)
      0
      sage: a(2)
      1
      sage: a(10)
      25
      sage: a.list(12)
      [0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30]

  Python

      >>> from sage.all import *
      >>> a = sloane.A002620;a
      Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      0
      >>> a(Integer(2))
      1
      >>> a(Integer(10))
      25
      >>> a.list(Integer(12))
      [0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A002808

  Bases: `SloaneSequence`

  The composite numbers: numbers \(n\) of the form \(xy\) for \(x > 1\) and \(y > 1\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A002808;a
      The composite numbers: numbers n of the form x*y for x > 1 and y > 1.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      6
      sage: a(11)
      20
      sage: a.list(12)
      [4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21]

  Python

      >>> from sage.all import *
      >>> a = sloane.A002808;a
      The composite numbers: numbers n of the form x*y for x > 1 and y > 1.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(2))
      6
      >>> a(Integer(11))
      20
      >>> a.list(Integer(12))
      [4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21]

  AUTHORS:

  - Jaap Spies (2007-01-26)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A002808.list(10)
        [4, 6, 8, 9, 10, 12, 14, 15, 16, 18]

    Python

        >>> from sage.all import *
        >>> sloane.A002808.list(Integer(10))
        [4, 6, 8, 9, 10, 12, 14, 15, 16, 18]

class sage.combinat.sloane_functions.A003418

  Bases: `SloaneSequence`

  Least common multiple (or lcm) of \(\{1, 2, \ldots, n\}\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A003418;a
      Least common multiple (or lcm) of {1, 2, ..., n}.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(13)
      360360
      sage: a.list(14)
      [1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360]
      sage: a(20.0)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  Python

      >>> from sage.all import *
      >>> a = sloane.A003418;a
      Least common multiple (or lcm) of {1, 2, ..., n}.
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      1
      >>> a(Integer(13))
      360360
      >>> a.list(Integer(14))
      [1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360]
      >>> a(RealNumber('20.0'))
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  AUTHOR:

  - Jaap Spies (2007-01-31)

class sage.combinat.sloane_functions.A004086

  Bases: `SloaneSequence`

  Read n backwards (referred to as \(R(n)\) in many sequences).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A004086;a
      Read n backwards (referred to as R(n) in many sequences).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(3333)
      3333
      sage: a(12345)
      54321
      sage: a.list(13)
      [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21]

  Python

      >>> from sage.all import *
      >>> a = sloane.A004086;a
      Read n backwards (referred to as R(n) in many sequences).
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      2
      >>> a(Integer(3333))
      3333
      >>> a(Integer(12345))
      54321
      >>> a.list(Integer(13))
      [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21]

  AUTHORS:

  - Jaap Spies (2007-02-02)

class sage.combinat.sloane_functions.A004526

  Bases: `SloaneSequence`

  The nonnegative integers repeated.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A004526;a
      The nonnegative integers repeated.
      sage: a(0)
      0
      sage: a(1)
      0
      sage: a(2)
      1
      sage: a(10)
      5
      sage: a.list(12)
      [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5]

  Python

      >>> from sage.all import *
      >>> a = sloane.A004526;a
      The nonnegative integers repeated.
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      0
      >>> a(Integer(2))
      1
      >>> a(Integer(10))
      5
      >>> a.list(Integer(12))
      [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A005100

  Bases: `SloaneSequence`

  Deficient numbers: \(\sigma(n) < 2n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A005100;a
      Deficient numbers: sigma(n) < 2n
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(12)
      14
      sage: a.list(12)
      [1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14]

  Python

      >>> from sage.all import *
      >>> a = sloane.A005100;a
      Deficient numbers: sigma(n) < 2n
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      2
      >>> a(Integer(12))
      14
      >>> a.list(Integer(12))
      [1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14]

  AUTHORS:

  - Jaap Spies (2007-01-26)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A005100.list(10)
        [1, 2, 3, 4, 5, 7, 8, 9, 10, 11]

    Python

        >>> from sage.all import *
        >>> sloane.A005100.list(Integer(10))
        [1, 2, 3, 4, 5, 7, 8, 9, 10, 11]

class sage.combinat.sloane_functions.A005101

  Bases: `SloaneSequence`

  Abundant numbers (sum of divisors of \(n\) exceeds \(2n\)).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A005101;a
      Abundant numbers (sum of divisors of n exceeds 2n).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      12
      sage: a(2)
      18
      sage: a(12)
      60
      sage: a.list(12)
      [12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60]

  Python

      >>> from sage.all import *
      >>> a = sloane.A005101;a
      Abundant numbers (sum of divisors of n exceeds 2n).
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      12
      >>> a(Integer(2))
      18
      >>> a(Integer(12))
      60
      >>> a.list(Integer(12))
      [12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60]

  AUTHORS:

  - Jaap Spies (2007-01-26)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A005101.list(10)
        [12, 18, 20, 24, 30, 36, 40, 42, 48, 54]

    Python

        >>> from sage.all import *
        >>> sloane.A005101.list(Integer(10))
        [12, 18, 20, 24, 30, 36, 40, 42, 48, 54]

class sage.combinat.sloane_functions.A005117

  Bases: `SloaneSequence`

  Square-free numbers.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A005117;a
      Square-free numbers.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      2
      sage: a(12)
      17
      sage: a.list(12)
      [1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17]

  Python

      >>> from sage.all import *
      >>> a = sloane.A005117;a
      Square-free numbers.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(2))
      2
      >>> a(Integer(12))
      17
      >>> a.list(Integer(12))
      [1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17]

  AUTHORS:

  - Jaap Spies (2007-01-25)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A005117.list(10)
        [1, 2, 3, 5, 6, 7, 10, 11, 13, 14]

    Python

        >>> from sage.all import *
        >>> sloane.A005117.list(Integer(10))
        [1, 2, 3, 5, 6, 7, 10, 11, 13, 14]

class sage.combinat.sloane_functions.A005408

  Bases: `SloaneSequence`

  The odd numbers a(n) = 2n + 1.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A005408;a
      The odd numbers a(n) = 2n + 1.
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      1
      sage: a(4)
      9
      sage: a(11)
      23
      sage: a.list(12)
      [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]

  Python

      >>> from sage.all import *
      >>> a = sloane.A005408;a
      The odd numbers a(n) = 2n + 1.
      >>> a(-Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      >>> a(Integer(0))
      1
      >>> a(Integer(4))
      9
      >>> a(Integer(11))
      23
      >>> a.list(Integer(12))
      [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A005843

  Bases: `SloaneSequence`

  The even numbers: \(a(n) = 2n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A005843;a
      The even numbers: a(n) = 2n.
      sage: a(0.0)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer
      sage: a(1)
      2
      sage: a(2)
      4
      sage: a(9)
      18
      sage: a.list(10)
      [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]

  Python

      >>> from sage.all import *
      >>> a = sloane.A005843;a
      The even numbers: a(n) = 2n.
      >>> a(RealNumber('0.0'))
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer
      >>> a(Integer(1))
      2
      >>> a(Integer(2))
      4
      >>> a(Integer(9))
      18
      >>> a.list(Integer(10))
      [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]

  AUTHORS:

  - Jaap Spies (2007-02-03)

class sage.combinat.sloane_functions.A006318

  Bases: `SloaneSequence`

  Large Schroeder numbers.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A006318;a
      Large Schroeder numbers.
      sage: a(0)
      1
      sage: a(1)
      2
      sage: a(2)
      6
      sage: a(9)
      206098
      sage: a.list(10)
      [1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098]

  Python

      >>> from sage.all import *
      >>> a = sloane.A006318;a
      Large Schroeder numbers.
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      2
      >>> a(Integer(2))
      6
      >>> a(Integer(9))
      206098
      >>> a.list(Integer(10))
      [1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098]

  AUTHORS:

  - Jaap Spies (2007-02-03)

class sage.combinat.sloane_functions.A006530

  Bases: `SloaneSequence`

  Largest prime dividing \(n\) (with \(a(1)=1\)).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A006530;a
      Largest prime dividing n (with a(1)=1).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(8)
      2
      sage: a(11)
      11
      sage: a.list(15)
      [1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5]

  Python

      >>> from sage.all import *
      >>> a = sloane.A006530;a
      Largest prime dividing n (with a(1)=1).
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      2
      >>> a(Integer(8))
      2
      >>> a(Integer(11))
      11
      >>> a.list(Integer(15))
      [1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A006882

  Bases: `SloaneSequence`

  Double factorials \(n!!\): \(a(n)=n \cdot a(n-2)\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A006882;a
      Double factorials n!!: a(n)=n*a(n-2).
      sage: a(0)
      1
      sage: a(2)
      2
      sage: a(8)
      384
      sage: a(20)
      3715891200
      sage: a.list(9)
      [1, 1, 2, 3, 8, 15, 48, 105, 384]

  Python

      >>> from sage.all import *
      >>> a = sloane.A006882;a
      Double factorials n!!: a(n)=n*a(n-2).
      >>> a(Integer(0))
      1
      >>> a(Integer(2))
      2
      >>> a(Integer(8))
      384
      >>> a(Integer(20))
      3715891200
      >>> a.list(Integer(9))
      [1, 1, 2, 3, 8, 15, 48, 105, 384]

  AUTHORS:

  - Jaap Spies (2007-01-24)

  df()

    Double factorials n!!: a(n)=n*a(n-2).

    EXAMPLES:

    Sage

        sage: it = sloane.A006882.df()
        sage: [next(it) for i in range(10)]
        [1, 1, 2, 3, 8, 15, 48, 105, 384, 945]

    Python

        >>> from sage.all import *
        >>> it = sloane.A006882.df()
        >>> [next(it) for i in range(Integer(10))]
        [1, 1, 2, 3, 8, 15, 48, 105, 384, 945]

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A006882.list(10)
        [1, 1, 2, 3, 8, 15, 48, 105, 384, 945]

    Python

        >>> from sage.all import *
        >>> sloane.A006882.list(Integer(10))
        [1, 1, 2, 3, 8, 15, 48, 105, 384, 945]

class sage.combinat.sloane_functions.A007318

  Bases: `SloaneSequence`

  Pascal’s triangle read by rows: \(C(n,k) = \binom{n}{k} = \frac {n!} {(k!(n-k)!)}\), \(0 \le k \le n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A007318
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(13)
      4
      sage: a.list(15)
      [1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1]
      sage: a(100)
      715

  Python

      >>> from sage.all import *
      >>> a = sloane.A007318
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      1
      >>> a(Integer(13))
      4
      >>> a.list(Integer(15))
      [1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1]
      >>> a(Integer(100))
      715

  AUTHORS:

  - Jaap Spies (2007-01-31)

  keyword = ['nonn', 'tabl', 'nice', 'easy', 'core', 'triangle']

class sage.combinat.sloane_functions.A008275

  Bases: `SloaneSequence`

  Triangle of Stirling numbers of first kind, \(s(n,k)\), \(n \ge 1\), \(1 \le k \le n\).

  The unsigned numbers are also called Stirling cycle numbers:

  \(|s(n,k)|\) = number of permutations of \(n\) objects with exactly \(k\) cycles.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A008275;a
      Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      -1
      sage: a(3)
      1
      sage: a(11)
      24
      sage: a.list(12)
      [1, -1, 1, 2, -3, 1, -6, 11, -6, 1, 24, -50]

  Python

      >>> from sage.all import *
      >>> a = sloane.A008275;a
      Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      -1
      >>> a(Integer(3))
      1
      >>> a(Integer(11))
      24
      >>> a.list(Integer(12))
      [1, -1, 1, 2, -3, 1, -6, 11, -6, 1, 24, -50]

  AUTHORS:

  - Jaap Spies (2007-02-02)

  keyword = ['sign', 'tabl', 'nice', 'core', 'triangle']
  s(n, k)

    EXAMPLES:

    Sage

        sage: sloane.A008275.s(4,2)
        11
        sage: sloane.A008275.s(5,2)
        -50
        sage: sloane.A008275.s(5,3)
        35

    Python

        >>> from sage.all import *
        >>> sloane.A008275.s(Integer(4),Integer(2))
        11
        >>> sloane.A008275.s(Integer(5),Integer(2))
        -50
        >>> sloane.A008275.s(Integer(5),Integer(3))
        35

class sage.combinat.sloane_functions.A008277

  Bases: `SloaneSequence`

  Triangle of Stirling numbers of 2nd kind, \(S2(n,k)\), \(n \ge 1\), \(1 \le k \le n\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A008277;a
      Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      1
      sage: a(3)
      1
      sage: a(4.0)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer
      sage: a.list(15)
      [1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1]

  Python

      >>> from sage.all import *
      >>> a = sloane.A008277;a
      Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      1
      >>> a(Integer(3))
      1
      >>> a(RealNumber('4.0'))
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer
      >>> a.list(Integer(15))
      [1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1]

  AUTHORS:

  - Jaap Spies (2007-01-31)

  keyword = ['nonn', 'tabl', 'nice', 'core', 'triangle']
  s2(n, k)

    Return the Stirling number S2(n,k) of the 2nd kind.

    EXAMPLES:

    Sage

        sage: sloane.A008277.s2(4,2)
        7

    Python

        >>> from sage.all import *
        >>> sloane.A008277.s2(Integer(4),Integer(2))
        7

class sage.combinat.sloane_functions.A008683

  Bases: `SloaneSequence`

  Möbius function \(\mu(n)\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A008683;a
      Moebius function mu(n).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      -1
      sage: a(12)
      0
      sage: a.list(12)
      [1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0]

  Python

      >>> from sage.all import *
      >>> a = sloane.A008683;a
      Moebius function mu(n).
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(2))
      -1
      >>> a(Integer(12))
      0
      >>> a.list(Integer(12))
      [1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A010060

  Bases: `SloaneSequence`

  Thue-Morse sequence.

  Let \(A_k\) denote the first \(2^k\) terms; then \(A_0 = 0\), and for \(k \ge 0\), \(A_{k+1} = A_k B_k\), where \(B_k\) is obtained from \(A_k\) by interchanging 0s and 1s.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A010060;a
      Thue-Morse sequence.
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      1
      sage: a(12)
      0
      sage: a.list(13)
      [0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0]

  Python

      >>> from sage.all import *
      >>> a = sloane.A010060;a
      Thue-Morse sequence.
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      1
      >>> a(Integer(12))
      0
      >>> a.list(Integer(13))
      [0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0]

  AUTHORS:

  - Jaap Spies (2007-02-02)

class sage.combinat.sloane_functions.A015521

  Bases: `RecurrenceSequence2`

  Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 3 a(n-1) + 4 a(n-2)\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A015521; a
      Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(8)
      13107
      sage: a(41)
      967140655691703339764941
      sage: a.list(12)
      [0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861]

  Python

      >>> from sage.all import *
      >>> a = sloane.A015521; a
      Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2).
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(8))
      13107
      >>> a(Integer(41))
      967140655691703339764941
      >>> a.list(Integer(12))
      [0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A015523

  Bases: `RecurrenceSequence2`

  Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 3 a(n-1) + 5 a(n-2)\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A015523; a
      Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(8)
      17727
      sage: a(41)
      6173719566474529739091481
      sage: a.list(12)
      [0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469]

  Python

      >>> from sage.all import *
      >>> a = sloane.A015523; a
      Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2).
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(8))
      17727
      >>> a(Integer(41))
      6173719566474529739091481
      >>> a.list(Integer(12))
      [0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A015530

  Bases: `RecurrenceSequence2`

  Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 4 a(n-1) + 3 a(n-2)\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A015530;a
      Linear 2nd order recurrence, a(n) = 4 a(n-1) + 3 a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      4
      sage: a.offset
      0
      sage: a(8)
      41008
      sage: a.list(9)
      [0, 1, 4, 19, 88, 409, 1900, 8827, 41008]

  Python

      >>> from sage.all import *
      >>> a = sloane.A015530;a
      Linear 2nd order recurrence, a(n) = 4 a(n-1) + 3 a(n-2).
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      4
      >>> a.offset
      0
      >>> a(Integer(8))
      41008
      >>> a.list(Integer(9))
      [0, 1, 4, 19, 88, 409, 1900, 8827, 41008]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A015531

  Bases: `RecurrenceSequence2`

  Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 4 a(n-1) + 5 a(n-2)\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A015531;a
      Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      4
      sage: a.offset
      0
      sage: a(8)
      65104
      sage: a(60)
      144560289664733924534327040115992228190104
      sage: a.list(9)
      [0, 1, 4, 21, 104, 521, 2604, 13021, 65104]

  Python

      >>> from sage.all import *
      >>> a = sloane.A015531;a
      Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2).
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      4
      >>> a.offset
      0
      >>> a(Integer(8))
      65104
      >>> a(Integer(60))
      144560289664733924534327040115992228190104
      >>> a.list(Integer(9))
      [0, 1, 4, 21, 104, 521, 2604, 13021, 65104]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A015551

  Bases: `RecurrenceSequence2`

  Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 6 a(n-1) + 5 a(n-2)\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A015551;a
      Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      6
      sage: a.offset
      0
      sage: a(8)
      570216
      sage: a(60)
      7110606606530059736761484557155863822531970573036
      sage: a.list(9)
      [0, 1, 6, 41, 276, 1861, 12546, 84581, 570216]

  Python

      >>> from sage.all import *
      >>> a = sloane.A015551;a
      Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2).
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      6
      >>> a.offset
      0
      >>> a(Integer(8))
      570216
      >>> a(Integer(60))
      7110606606530059736761484557155863822531970573036
      >>> a.list(Integer(9))
      [0, 1, 6, 41, 276, 1861, 12546, 84581, 570216]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A018252

  Bases: `SloaneSequence`

  The nonprime numbers, starting with 1.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A018252;a
      The nonprime numbers.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      4
      sage: a(9)
      15
      sage: a.list(10)
      [1, 4, 6, 8, 9, 10, 12, 14, 15, 16]

  Python

      >>> from sage.all import *
      >>> a = sloane.A018252;a
      The nonprime numbers.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      4
      >>> a(Integer(9))
      15
      >>> a.list(Integer(10))
      [1, 4, 6, 8, 9, 10, 12, 14, 15, 16]

  AUTHORS:

  - Jaap Spies (2007-02-04)

class sage.combinat.sloane_functions.A020639

  Bases: `SloaneSequence`

  Least prime dividing \(n\) with \(a(1)=1\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A020639;a
      Least prime dividing n (a(1)=1).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(13)
      13
      sage: a.list(14)
      [1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2]

  Python

      >>> from sage.all import *
      >>> a = sloane.A020639;a
      Least prime dividing n (a(1)=1).
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      1
      >>> a(Integer(13))
      13
      >>> a.list(Integer(14))
      [1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2]

  AUTHORS:

  - Jaap Spies (2007-01-25)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A020639.list(10)
        [1, 2, 3, 2, 5, 2, 7, 2, 3, 2]

    Python

        >>> from sage.all import *
        >>> sloane.A020639.list(Integer(10))
        [1, 2, 3, 2, 5, 2, 7, 2, 3, 2]

class sage.combinat.sloane_functions.A046660(offset=1)

  Bases: `SloaneSequence`

  Excess of \(n\) = number of prime divisors (with multiplicity) - number of prime divisors (without multiplicity).

  \(\Omega(n) - \omega(n)\).

  INPUT:

  - `n` – positive integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A046660; a
      Excess of n = Bigomega (with multiplicity) - omega (without multiplicity).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      0
      sage: a(8)
      2
      sage: a(41)
      0
      sage: a(84792)
      2
      sage: a.list(12)
      [0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1]

  Python

      >>> from sage.all import *
      >>> a = sloane.A046660; a
      Excess of n = Bigomega (with multiplicity) - omega (without multiplicity).
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      0
      >>> a(Integer(8))
      2
      >>> a(Integer(41))
      0
      >>> a(Integer(84792))
      2
      >>> a.list(Integer(12))
      [0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1]

  AUTHORS:

  > - Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A049310

  Bases: `SloaneSequence`

  Triangle of coefficients of Chebyshev’s \(S(n,x)\): \(U(n, \frac x 2)\) polynomials (exponents in increasing order).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A049310;a
      Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order).
      sage: a(0)
      1
      sage: a(1)
      0
      sage: a(13)
      0
      sage: a.list(15)
      [1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1]
      sage: a(200)
      0
      sage: a.keyword
      ['sign', 'tabl', 'nice', 'easy', 'core', 'triangle']

  Python

      >>> from sage.all import *
      >>> a = sloane.A049310;a
      Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order).
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      0
      >>> a(Integer(13))
      0
      >>> a.list(Integer(15))
      [1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1]
      >>> a(Integer(200))
      0
      >>> a.keyword
      ['sign', 'tabl', 'nice', 'easy', 'core', 'triangle']

  AUTHORS:

  - Jaap Spies (2007-01-31)

  keyword = ['sign', 'tabl', 'nice', 'easy', 'core', 'triangle']

class sage.combinat.sloane_functions.A051959

  Bases: `RecurrenceSequence`

  Linear second order recurrence. A051959.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A051959; a
      Linear second order recurrence. A051959.
      sage: a(0)
      1
      sage: a(1)
      10
      sage: a(8)
      9969
      sage: a(41)
      42834431872413650
      sage: a.list(12)
      [1, 10, 36, 104, 273, 686, 1688, 4112, 9969, 24114, 58268, 140728]

  Python

      >>> from sage.all import *
      >>> a = sloane.A051959; a
      Linear second order recurrence. A051959.
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      10
      >>> a(Integer(8))
      9969
      >>> a(Integer(41))
      42834431872413650
      >>> a.list(Integer(12))
      [1, 10, 36, 104, 273, 686, 1688, 4112, 9969, 24114, 58268, 140728]

  AUTHORS:

  - Jaap Spies (2007-01-19)

  g(k)

    EXAMPLES:

    Sage

        sage: sloane.A051959.g(2)
        15
        sage: sloane.A051959.g(1)
        0

    Python

        >>> from sage.all import *
        >>> sloane.A051959.g(Integer(2))
        15
        >>> sloane.A051959.g(Integer(1))
        0

class sage.combinat.sloane_functions.A055790

  Bases: `ExtremesOfPermanentsSequence2`

  \(a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2]\).

  With offset 1, permanent of (0,1)-matrix of size n X (n+d) with d=1 and n-1 zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A055790;a
      a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2].
      sage: a(0)
      0
      sage: a(1)
      2
      sage: a(2)
      4
      sage: a.offset
      0
      sage: a(8)
      165016
      sage: a(22)
      10356214297533070441564
      sage: a.list(9)
      [0, 2, 4, 14, 64, 362, 2428, 18806, 165016]

  Python

      >>> from sage.all import *
      >>> a = sloane.A055790;a
      a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2].
      >>> a(Integer(0))
      0
      >>> a(Integer(1))
      2
      >>> a(Integer(2))
      4
      >>> a.offset
      0
      >>> a(Integer(8))
      165016
      >>> a(Integer(22))
      10356214297533070441564
      >>> a.list(Integer(9))
      [0, 2, 4, 14, 64, 362, 2428, 18806, 165016]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A061084

  Bases: `SloaneSequence`

  Fibonacci-type sequence based on subtraction: \(a(0) = 1\), \(a(1) = 2\) and \(a(n) = a(n-2)-a(n-1)\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A061084; a
      Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).
      sage: a(0)
      1
      sage: a(1)
      2
      sage: a(8)
      -29
      sage: a(22)
      -24476
      sage: a.list(12)
      [1, 2, -1, 3, -4, 7, -11, 18, -29, 47, -76, 123]
      sage: a.keyword
      ['sign', 'easy', 'nice']

  Python

      >>> from sage.all import *
      >>> a = sloane.A061084; a
      Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).
      >>> a(Integer(0))
      1
      >>> a(Integer(1))
      2
      >>> a(Integer(8))
      -29
      >>> a(Integer(22))
      -24476
      >>> a.list(Integer(12))
      [1, 2, -1, 3, -4, 7, -11, 18, -29, 47, -76, 123]
      >>> a.keyword
      ['sign', 'easy', 'nice']

  AUTHORS:

  - Jaap Spies (2007-01-18)

  keyword = ['sign', 'easy', 'nice']

class sage.combinat.sloane_functions.A064553

  Bases: `SloaneSequence`

  \(a(1) = 1\), \(a(prime(i)) = i + 1\) for \(i > 0\) and \(a(u \cdot v) = a(u) \cdot a(v)\) for \(u, v > 0\).

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A064553;a
      a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(9)
      9
      sage: a.list(16)
      [1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16]

  Python

      >>> from sage.all import *
      >>> a = sloane.A064553;a
      a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      1
      >>> a(Integer(2))
      2
      >>> a(Integer(9))
      9
      >>> a.list(Integer(16))
      [1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16]

  AUTHORS:

  - Jaap Spies (2007-02-04)

class sage.combinat.sloane_functions.A079922(offset=1)

  Bases: `SloaneSequence`

  Function returns solutions to the Dancing School problem with \(n\) girls and \(n+3\) boys.

  The value is \(per(B)\), the permanent of the (0,1)-matrix \(B\) of size \(n \times n+3\) with \(b(i,j)=1\) if and only if \(i \le j \le i+n\).

  REFERENCES:

  - Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006

  INPUT:

  - `n` – positive integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A079922; a
      Solutions to the Dancing School problem with n girls and n+3 boys
      sage: a.offset
      1
      sage: a(1)
      4
      sage: a(8)
      2227
      sage: a.list(8)
      [4, 13, 36, 90, 212, 478, 1044, 2227]

  Python

      >>> from sage.all import *
      >>> a = sloane.A079922; a
      Solutions to the Dancing School problem with n girls and n+3 boys
      >>> a.offset
      1
      >>> a(Integer(1))
      4
      >>> a(Integer(8))
      2227
      >>> a.list(Integer(8))
      [4, 13, 36, 90, 212, 478, 1044, 2227]

  Compare: Searching Sloane’s online database… Solution to the Dancing School Problem with n girls and n+3 boys: f(n,3). [4, 13, 36, 90, 212, 478, 1044, 2227]

  Sage

      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be a positive integer

  [Python]

      >>> from sage.all import *
      >>> a(-Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be a positive integer

  AUTHORS:

  > - Jaap Spies (2007-01-14)

class sage.combinat.sloane_functions.A079923(offset=1)

  Bases: `SloaneSequence`

  Function returns solutions to the Dancing School problem with \(n\) girls and \(n+4\) boys.

  The value is \(per(B)\), the permanent of the (0,1)-matrix \(B\) of size \(n \times n+3\) with \(b(i,j)=1\) if and only if \(i \le j \le i+n\).

  REFERENCES:

  - Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006

  INPUT:

  - `n` – positive integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A079923; a
      Solutions to the Dancing School problem with n girls and n+4 boys
      sage: a.offset
      1
      sage: a(1)
      5
      sage: a(8)
      15458
      sage: a.list(8)
      [5, 21, 76, 246, 738, 2108, 5794, 15458]

  Python

      >>> from sage.all import *
      >>> a = sloane.A079923; a
      Solutions to the Dancing School problem with n girls and n+4 boys
      >>> a.offset
      1
      >>> a(Integer(1))
      5
      >>> a(Integer(8))
      15458
      >>> a.list(Integer(8))
      [5, 21, 76, 246, 738, 2108, 5794, 15458]

  Compare: Searching Sloane’s online database… Solution to the Dancing School Problem with n girls and n+4 boys: f(n,4). [5, 21, 76, 246, 738, 2108, 5794, 15458]

  Sage

      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  [Python]

      >>> from sage.all import *
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  AUTHORS:

  > - Jaap Spies (2007-01-17)

class sage.combinat.sloane_functions.A082411

  Bases: `RecurrenceSequence2`

  Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

  \(a(0) = 407389224418\), \(a(1) = 76343678551\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime, that R. L. Graham in 1964 stated did not contain any primes.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A082411;a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      sage: a(1)
      76343678551
      sage: a(2)
      483732902969
      sage: a(3)
      560076581520
      sage: a(20)
      2219759332689173
      sage: a.list(4)
      [407389224418, 76343678551, 483732902969, 560076581520]

  Python

      >>> from sage.all import *
      >>> a = sloane.A082411;a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      >>> a(Integer(1))
      76343678551
      >>> a(Integer(2))
      483732902969
      >>> a(Integer(3))
      560076581520
      >>> a(Integer(20))
      2219759332689173
      >>> a.list(Integer(4))
      [407389224418, 76343678551, 483732902969, 560076581520]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A083103

  Bases: `RecurrenceSequence2`

  Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

  \(a(0) = 1786772701928802632268715130455793\), \(a(1) = 1059683225053915111058165141686995\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co- prime, that R. L. Graham in 1964 stated did not contain any primes. It has not been verified. Graham made a mistake in the calculation that was corrected by D. E. Knuth in 1990.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A083103;a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      sage: a(1)
      1059683225053915111058165141686995
      sage: a(2)
      2846455926982717743326880272142788
      sage: a(3)
      3906139152036632854385045413829783
      sage: a.offset
      0
      sage: a(8)
      45481392851206651551714764671352204
      sage: a(20)
      14639253684254059531823985143948191708
      sage: a.list(4)
      [1786772701928802632268715130455793, 1059683225053915111058165141686995, 2846455926982717743326880272142788, 3906139152036632854385045413829783]

  Python

      >>> from sage.all import *
      >>> a = sloane.A083103;a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      >>> a(Integer(1))
      1059683225053915111058165141686995
      >>> a(Integer(2))
      2846455926982717743326880272142788
      >>> a(Integer(3))
      3906139152036632854385045413829783
      >>> a.offset
      0
      >>> a(Integer(8))
      45481392851206651551714764671352204
      >>> a(Integer(20))
      14639253684254059531823985143948191708
      >>> a.list(Integer(4))
      [1786772701928802632268715130455793, 1059683225053915111058165141686995, 2846455926982717743326880272142788, 3906139152036632854385045413829783]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A083104

  Bases: `RecurrenceSequence2`

  Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

  \(a(0) = 331635635998274737472200656430763\), \(a(1) = 1510028911088401971189590305498785\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime. It was found by Ronald Graham in 1990.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A083104;a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      sage: a(3)
      3351693458175078679851381267428333
      sage: a.offset
      0
      sage: a(8)
      36021870400834012982120004949074404
      sage: a(20)
      11601914177621826012468849361236300628

  Python

      >>> from sage.all import *
      >>> a = sloane.A083104;a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      >>> a(Integer(3))
      3351693458175078679851381267428333
      >>> a.offset
      0
      >>> a(Integer(8))
      36021870400834012982120004949074404
      >>> a(Integer(20))
      11601914177621826012468849361236300628

  AUTHORS:

  - Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A083105

  Bases: `RecurrenceSequence2`

  Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

  \(a(0) = 62638280004239857\), \(a(1) = 49463435743205655\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime. It was found by Donald Knuth in 1990.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A083105;a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      sage: a(1)
      49463435743205655
      sage: a(2)
      112101715747445512
      sage: a(3)
      161565151490651167
      sage: a.offset
      0
      sage: a(8)
      1853029790662436896
      sage: a(20)
      596510791500513098192
      sage: a.list(4)
      [62638280004239857, 49463435743205655, 112101715747445512, 161565151490651167]

  Python

      >>> from sage.all import *
      >>> a = sloane.A083105;a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      >>> a(Integer(1))
      49463435743205655
      >>> a(Integer(2))
      112101715747445512
      >>> a(Integer(3))
      161565151490651167
      >>> a.offset
      0
      >>> a(Integer(8))
      1853029790662436896
      >>> a(Integer(20))
      596510791500513098192
      >>> a.list(Integer(4))
      [62638280004239857, 49463435743205655, 112101715747445512, 161565151490651167]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A083216

  Bases: `RecurrenceSequence2`

  Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

  \(a(0) = 20615674205555510\), \(a(1) = 3794765361567513\). This is a second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime that does not contain any primes. It was found by Herbert Wilf in 1990.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A083216; a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      sage: a(0)
      20615674205555510
      sage: a(1)
      3794765361567513
      sage: a(8)
      347693837265139403
      sage: a(41)
      2738025383211084205003383
      sage: a.list(4)
      [20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536]

  Python

      >>> from sage.all import *
      >>> a = sloane.A083216; a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      >>> a(Integer(0))
      20615674205555510
      >>> a(Integer(1))
      3794765361567513
      >>> a(Integer(8))
      347693837265139403
      >>> a(Integer(41))
      2738025383211084205003383
      >>> a.list(Integer(4))
      [20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A090010

  Bases: `ExtremesOfPermanentsSequence2`

  Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=6\) and \(n\) zeros not on a line.

  ` a(n) = (n+5)*a(n-1) + (n-1)*a(n-2), a(1)=6, a(2)=43`.

  This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A090010;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      6
      sage: a(2)
      43
      sage: a.offset
      1
      sage: a(8)
      67741129
      sage: a(22)
      192416593029158989003270143
      sage: a.list(9)
      [6, 43, 356, 3333, 34754, 398959, 4996032, 67741129, 988344062]

  Python

      >>> from sage.all import *
      >>> a = sloane.A090010;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      6
      >>> a(Integer(2))
      43
      >>> a.offset
      1
      >>> a(Integer(8))
      67741129
      >>> a(Integer(22))
      192416593029158989003270143
      >>> a.list(Integer(9))
      [6, 43, 356, 3333, 34754, 398959, 4996032, 67741129, 988344062]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A090012

  Bases: `SloaneSequence`

  Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=2\) and \(n-1\) zeros not on a line.

  \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2)\), \(a(1)=3\) and \(a(2)=9\)

  This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A090012;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      3
      sage: a(2)
      9
      sage: a.offset
      1
      sage: a(8)
      890901
      sage: a(22)
      129020386652297208795129
      sage: a.list(9)
      [3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387]

  Python

      >>> from sage.all import *
      >>> a = sloane.A090012;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      3
      >>> a(Integer(2))
      9
      >>> a.offset
      1
      >>> a(Integer(8))
      890901
      >>> a(Integer(22))
      129020386652297208795129
      >>> a.list(Integer(9))
      [3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A090013

  Bases: `SloaneSequence`

  Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=3\) and \(n-1\) zeros not on a line.

  \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=4, a(2)=16]\)

  This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A090013;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      4
      sage: a(2)
      16
      sage: a.offset
      1
      sage: a(8)
      3481096
      sage: a(22)
      1112998577171142607670336
      sage: a.list(9)
      [4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284]

  Python

      >>> from sage.all import *
      >>> a = sloane.A090013;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      4
      >>> a(Integer(2))
      16
      >>> a.offset
      1
      >>> a(Integer(8))
      3481096
      >>> a(Integer(22))
      1112998577171142607670336
      >>> a.list(Integer(9))
      [4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A090014

  Bases: `SloaneSequence`

  Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=4\) and \(n-1\) zeros not on a line.

  \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=5, a(2)=25]\)

  This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A090014;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      5
      sage: a(2)
      25
      sage: a.offset
      1
      sage: a(8)
      11016595
      sage: a(22)
      7469733600354446865509725
      sage: a.list(9)
      [5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365]

  Python

      >>> from sage.all import *
      >>> a = sloane.A090014;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      5
      >>> a(Integer(2))
      25
      >>> a.offset
      1
      >>> a(Integer(8))
      11016595
      >>> a(Integer(22))
      7469733600354446865509725
      >>> a.list(Integer(9))
      [5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A090015

  Bases: `SloaneSequence`

  Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=5\) and \(n-1\) zeros not on a line.

  \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=6, a(2)=36]\)

  This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A090015;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      6
      sage: a(2)
      36
      sage: a.offset
      1
      sage: a(8)
      29976192
      sage: a(22)
      41552258517692116794936876
      sage: a.list(9)
      [6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]

  Python

      >>> from sage.all import *
      >>> a = sloane.A090015;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      6
      >>> a(Integer(2))
      36
      >>> a.offset
      1
      >>> a(Integer(8))
      29976192
      >>> a(Integer(22))
      41552258517692116794936876
      >>> a.list(Integer(9))
      [6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A090016

  Bases: `SloaneSequence`

  Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=6\) and \(n-1\) zeros not on a line.

  \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=7, a(2)=49]\)

  \(A090016 a(n) = A090010(n-1) + A090010(n), a(1)=7\)

  This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A090016;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      7
      sage: a(2)
      49
      sage: a.offset
      1
      sage: a(8)
      72737161
      sage: a(22)
      199341969448774341802426289
      sage: a.list(9)
      [7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191]

  Python

      >>> from sage.all import *
      >>> a = sloane.A090016;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(1))
      7
      >>> a(Integer(2))
      49
      >>> a.offset
      1
      >>> a(Integer(8))
      72737161
      >>> a(Integer(22))
      199341969448774341802426289
      >>> a.list(Integer(9))
      [7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A109814

  Bases: `SloaneSequence`

  The \(n\) th term of the sequence \(a(n)\) is the largest \(k\) such that \(n\) can be written as sum of \(k\) consecutive integers.

  By definition, \(n\) is the sum of at most \(a(n)\) consecutive positive integers. Suppose \(n\) is to be written as sum of \(k\) consecutive integers starting with \(m\), then \(2n = k(2m + k - 1)\). Only one of the factors is odd. For each odd divisor \(d\) of \(n\) there is a unique corresponding \(k = min(d,2n/d)\). \(a(n)\) can be alternatively defined as the largest among those \(k\) .

  See also

  - Wikipedia article Polite_number

  - An exercise sheet (with answers) about sums of consecutive integers.

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A109814; a
      a(n) is the largest k such that n can be written as sum of k consecutive positive integers.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      1
      sage: a.list(9)
      [1, 1, 2, 1, 2, 3, 2, 1, 3]

  Python

      >>> from sage.all import *
      >>> a = sloane.A109814; a
      a(n) is the largest k such that n can be written as sum of k consecutive positive integers.
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(2))
      1
      >>> a.list(Integer(9))
      [1, 1, 2, 1, 2, 3, 2, 1, 3]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A111774

  Bases: `SloaneSequence`

  Sequence of numbers of the third kind, i.e., numbers that can be written as a sum of at least three consecutive positive integers.

  Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of \(k\) consecutive integers (other than the trivial \(n = n\) for \(k = 1\)).

  See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A111774; a
      Numbers that can be written as a sum of at least three consecutive positive integers.
      sage: a(1)
      6
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(100)
      141
      sage: a(156)
      209
      sage: a(302)
      386
      sage: a.list(12)
      [6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
      sage: a(1/3)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  Python

      >>> from sage.all import *
      >>> a = sloane.A111774; a
      Numbers that can be written as a sum of at least three consecutive positive integers.
      >>> a(Integer(1))
      6
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(100))
      141
      >>> a(Integer(156))
      209
      >>> a(Integer(302))
      386
      >>> a.list(Integer(12))
      [6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
      >>> a(Integer(1)/Integer(3))
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  AUTHORS:

  - Jaap Spies (2007-01-13)

  is_number_of_the_third_kind(n)

    Return `True` if and only if \(n\) is a number of the third kind.

    A number is of the third kind if it can be written as a sum of at least three consecutive positive integers. Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of \(k\) consecutive integers (other than the trivial \(n = n\) for \(k = 1\)).

    See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

    INPUT:

    - `n` – positive integer

    OUTPUT: `True` if \(n\) is not prime and not a power of 2

    EXAMPLES:

    Sage

        sage: a = sloane.A111774
        sage: a.is_number_of_the_third_kind(6)
        True
        sage: a.is_number_of_the_third_kind(100)
        True
        sage: a.is_number_of_the_third_kind(16)
        False
        sage: a.is_number_of_the_third_kind(97)
        False

    Python

        >>> from sage.all import *
        >>> a = sloane.A111774
        >>> a.is_number_of_the_third_kind(Integer(6))
        True
        >>> a.is_number_of_the_third_kind(Integer(100))
        True
        >>> a.is_number_of_the_third_kind(Integer(16))
        False
        >>> a.is_number_of_the_third_kind(Integer(97))
        False

    AUTHORS:

    - Jaap Spies (2006-12-09)

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A111774.list(12)
        [6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]

    Python

        >>> from sage.all import *
        >>> sloane.A111774.list(Integer(12))
        [6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]

class sage.combinat.sloane_functions.A111775

  Bases: `SloaneSequence`

  Number of ways \(n\) can be written as a sum of at least three consecutive integers.

  Powers of 2 and (odd) primes can not be written as a sum of at least three consecutive integers. \(a(n)\) strongly depends on the number of odd divisors of \(n\) (A001227): Suppose \(n\) is to be written as sum of \(k\) consecutive integers starting with \(m\), then \(2n = k(2m + k - 1)\). Only one of the factors is odd. For each odd divisor of \(n\) there is a unique corresponding \(k\), \(k=1\) and \(k=2\) must be excluded.

  See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A111775; a
      Number of ways n can be written as a sum of at least three consecutive integers.

  Python

      >>> from sage.all import *
      >>> a = sloane.A111775; a
      Number of ways n can be written as a sum of at least three consecutive integers.

  Sage

      sage: a(1)
      0
      sage: a(0)
      0

  [Python]

      >>> from sage.all import *
      >>> a(Integer(1))
      0
      >>> a(Integer(0))
      0

  We have a(15)=2 because 15 = 4+5+6 and 15 = 1+2+3+4+5. The number of odd divisors of 15 is 4.

  Sage

      sage: a(15)
      2

  Python

      >>> from sage.all import *
      >>> a(Integer(15))
      2

  Sage

      sage: a(100)
      2
      sage: a(256)
      0
      sage: a(29)
      0
      sage: a.list(20)
      [0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0]
      sage: a(1/3)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  [Python]

      >>> from sage.all import *
      >>> a(Integer(100))
      2
      >>> a(Integer(256))
      0
      >>> a(Integer(29))
      0
      >>> a.list(Integer(20))
      [0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0]
      >>> a(Integer(1)/Integer(3))
      Traceback (most recent call last):
      ...
      TypeError: input must be an int or Integer

  AUTHORS:

  - Jaap Spies (2006-12-09)

class sage.combinat.sloane_functions.A111787

  Bases: `SloaneSequence`

  This function returns the \(n\)-th number of Sloane’s sequence A111787

  \(a(n)=0\) if \(n\) is an odd prime or a power of 2. For numbers of the third kind (see A111774) we proceed as follows: suppose \(n\) is to be written as sum of \(k\) consecutive integers starting with \(m\), then \(2n = k(2m + k - 1)\). Let \(p\) be the smallest odd prime divisor of \(n\) then \(a(n) = min(p,2n/p)\).

  See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

  INPUT:

  - `n` – nonnegative integer

  OUTPUT: integer; function value

  EXAMPLES:

  Sage

      sage: a = sloane.A111787; a
      a(n) is the least k >= 3 such that n can be written as sum of k consecutive integers. a(n)=0 if such a k does not exist.
      sage: a.offset
      1
      sage: a(1)
      0
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(100)
      5
      sage: a(256)
      0
      sage: a(29)
      0
      sage: a.list(20)
      [0, 0, 0, 0, 0, 3, 0, 0, 3, 4, 0, 3, 0, 4, 3, 0, 0, 3, 0, 5]
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be a positive integer

  Python

      >>> from sage.all import *
      >>> a = sloane.A111787; a
      a(n) is the least k >= 3 such that n can be written as sum of k consecutive integers. a(n)=0 if such a k does not exist.
      >>> a.offset
      1
      >>> a(Integer(1))
      0
      >>> a(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      >>> a(Integer(100))
      5
      >>> a(Integer(256))
      0
      >>> a(Integer(29))
      0
      >>> a.list(Integer(20))
      [0, 0, 0, 0, 0, 3, 0, 0, 3, 4, 0, 3, 0, 4, 3, 0, 0, 3, 0, 5]
      >>> a(-Integer(1))
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be a positive integer

  AUTHORS:

  - Jaap Spies (2007-01-14)

class sage.combinat.sloane_functions.ExponentialNumbers(a)

  Bases: `SloaneSequence`

  A sequence of Exponential numbers.

  EXAMPLES:

  Sage

      sage: from sage.combinat.sloane_functions import ExponentialNumbers
      sage: ExponentialNumbers(0)
      Sequence of Exponential numbers around 0

  Python

      >>> from sage.all import *
      >>> from sage.combinat.sloane_functions import ExponentialNumbers
      >>> ExponentialNumbers(Integer(0))
      Sequence of Exponential numbers around 0

class sage.combinat.sloane_functions.ExtremesOfPermanentsSequence(offset=1)

  Bases: `SloaneSequence`

  gen(a0, a1, d)

    EXAMPLES:

    Sage

        sage: it = sloane.A000153.gen(0,1,2)
        sage: [next(it) for i in range(5)]
        [0, 1, 2, 7, 32]

    Python

        >>> from sage.all import *
        >>> it = sloane.A000153.gen(Integer(0),Integer(1),Integer(2))
        >>> [next(it) for i in range(Integer(5))]
        [0, 1, 2, 7, 32]

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A000153.list(8)
        [0, 1, 2, 7, 32, 181, 1214, 9403]

    Python

        >>> from sage.all import *
        >>> sloane.A000153.list(Integer(8))
        [0, 1, 2, 7, 32, 181, 1214, 9403]

class sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2(offset=1)

  Bases: `ExtremesOfPermanentsSequence`

  gen(a0, a1, d)

    EXAMPLES:

    Sage

        sage: from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2
        sage: e = ExtremesOfPermanentsSequence2()
        sage: it = e.gen(6,43,6)
        sage: [next(it) for i in range(5)]
        [6, 43, 307, 2542, 23799]

    Python

        >>> from sage.all import *
        >>> from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2
        >>> e = ExtremesOfPermanentsSequence2()
        >>> it = e.gen(Integer(6),Integer(43),Integer(6))
        >>> [next(it) for i in range(Integer(5))]
        [6, 43, 307, 2542, 23799]

class sage.combinat.sloane_functions.RecurrenceSequence(offset=1)

  Bases: `SloaneSequence`

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A001110.list(8)
        [0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]

    Python

        >>> from sage.all import *
        >>> sloane.A001110.list(Integer(8))
        [0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]

class sage.combinat.sloane_functions.RecurrenceSequence2(offset=1)

  Bases: `SloaneSequence`

  list(n)

    EXAMPLES:

    Sage

        sage: sloane.A001906.list(10)
        [0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]

    Python

        >>> from sage.all import *
        >>> sloane.A001906.list(Integer(10))
        [0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]

class sage.combinat.sloane_functions.Sloane

  Bases: `SageObject`

  A collection of Sloane generating functions.

  This class inspects sage.combinat.sloane_functions, accumulating all the SloaneSequence classes starting with ‘A’. These are listed for tab completion, but not instantiated until requested.

  EXAMPLES:

  Ensure we have lots of entries:

  Sage

      sage: len(sloane.__dir__()) > 100
      True

  Python

      >>> from sage.all import *
      >>> len(sloane.__dir__()) > Integer(100)
      True

  Ensure none are being incorrectly returned:

  Sage

      sage: [ None for n in sloane.__dir__() if not n.startswith('A') ]
      []

  [Python]

      >>> from sage.all import *
      >>> [ None for n in sloane.__dir__() if not n.startswith('A') ]
      []

  Ensure we can access dynamic constructions and cache correctly:

  Sage

      sage: s = sloane.A000587
      sage: s is sloane.A000587
      True

  Python

      >>> from sage.all import *
      >>> s = sloane.A000587
      >>> s is sloane.A000587
      True

  Ensure that we can access other functions in parent classes:

  Sage

      sage: sloane.__class__
      <class 'sage.combinat.sloane_functions.Sloane'>

  [Python]

      >>> from sage.all import *
      >>> sloane.__class__
      <class 'sage.combinat.sloane_functions.Sloane'>

  AUTHORS:

  - Nick Alexander

class sage.combinat.sloane_functions.SloaneSequence(offset=1)

  Bases: `SageObject`

  Base class for a Sloane integer sequence.

  list(n)

    Return `n` terms of the sequence:

        sequence[offset], sequence[offset+1], ..., sequence[offset+n-1].

    EXAMPLES:

    Sage

        sage: sloane.A000012.list(4)
        [1, 1, 1, 1]

    Python

        >>> from sage.all import *
        >>> sloane.A000012.list(Integer(4))
        [1, 1, 1, 1]

sage.combinat.sloane_functions.perm_mh(m, h)

  This functions calculates \(f(g,h)\) from Sloane’s sequences A079908-A079928

  INPUT:

  - `m` – positive integer

  - `h` – nonnegative integer

  OUTPUT: permanent of the \(m \times (m+h)\) matrix, etc.

  EXAMPLES:

  Sage

      sage: from sage.combinat.sloane_functions import perm_mh
      sage: perm_mh(3,3)
      36
      sage: perm_mh(3,4)
      76

  Python

      >>> from sage.all import *
      >>> from sage.combinat.sloane_functions import perm_mh
      >>> perm_mh(Integer(3),Integer(3))
      36
      >>> perm_mh(Integer(3),Integer(4))
      76

  AUTHORS:

  - Jaap Spies (2006)

sage.combinat.sloane_functions.recur_gen2(a0, a1, a2, a3)

  Homogeneous general second-order linear recurrence generator with fixed coefficients.

  a(0) = a0, a(1) = a1, a(n) = a2*a(n-1) + a3*a(n-2)

  EXAMPLES:

  Sage

      sage: from sage.combinat.sloane_functions import recur_gen2
      sage: it = recur_gen2(1,1,1,1)
      sage: [next(it) for i in range(10)]
      [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

  Python

      >>> from sage.all import *
      >>> from sage.combinat.sloane_functions import recur_gen2
      >>> it = recur_gen2(Integer(1),Integer(1),Integer(1),Integer(1))
      >>> [next(it) for i in range(Integer(10))]
      [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

sage.combinat.sloane_functions.recur_gen2b(a0, a1, a2, a3, b)

  Inhomogeneous second-order linear recurrence generator with fixed coefficients and \(b = f(n)\)

  \(a(0) = a0\), \(a(1) = a1\), \(a(n) = a2*a(n-1) + a3*a(n-2) +f(n)\).

  EXAMPLES:

  Sage

      sage: from sage.combinat.sloane_functions import recur_gen2b
      sage: it = recur_gen2b(1,1,1,1, lambda n: 0)
      sage: [next(it) for i in range(10)]
      [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

  Python

      >>> from sage.all import *
      >>> from sage.combinat.sloane_functions import recur_gen2b
      >>> it = recur_gen2b(Integer(1),Integer(1),Integer(1),Integer(1), lambda n: Integer(0))
      >>> [next(it) for i in range(Integer(10))]
      [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

sage.combinat.sloane_functions.recur_gen3(a0, a1, a2, a3, a4, a5)

  Homogeneous general third-order linear recurrence generator with fixed coefficients

  a(0) = a0, a(1) = a1, a(2) = a2, a(n) = a3*a(n-1) + a4*a(n-2) + a5*a(n-3)

  EXAMPLES:

  Sage

      sage: from sage.combinat.sloane_functions import recur_gen3
      sage: it = recur_gen3(1,1,1,1,1,1)
      sage: [next(it) for i in range(10)]
      [1, 1, 1, 3, 5, 9, 17, 31, 57, 105]

  Python

      >>> from sage.all import *
      >>> from sage.combinat.sloane_functions import recur_gen3
      >>> it = recur_gen3(Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(1))
      >>> [next(it) for i in range(Integer(10))]
      [1, 1, 1, 3, 5, 9, 17, 31, 57, 105]