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: 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

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: a(0)
...
ValueError: input n (=0) must be a positive integer
sage: a(1/3)
...
TypeError: input must be an int, long, or Integer

You can also change how a sequence prints:

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.

TESTS:

sage: a = sloane.A000001;
sage: a == loads(dumps(a))
True

AUTHORS:

William Stein: framework
Jaap Spies: most sequences
Nick Alexander: updated framework

class sage.combinat.sloane_functions.A000001¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000004¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000005¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000007¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000008¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000009¶

Bases: sage.combinat.sloane_functions.SloaneSequence

cf()¶

EXAMPLES:

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

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A000010¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000012¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000015¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000016¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000027¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000030¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000032¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000035¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000040¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000041¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000043¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000045¶

Bases: sage.combinat.sloane_functions.SloaneSequence

fib()¶

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

EXAMPLES:

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

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A000069¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000073¶

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A000079¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000085¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000100¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000108¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000110¶: Bases: sage.combinat.sloane_functions.ExponentialNumbers

class sage.combinat.sloane_functions.A000120¶

Bases: sage.combinat.sloane_functions.SloaneSequence

f(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A000124¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000129¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A000142¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000153¶: Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence

class sage.combinat.sloane_functions.A000165¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000166¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000169¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000203¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000204¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000213¶

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A000217¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000225¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000244¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000255¶: Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence

class sage.combinat.sloane_functions.A000261¶: Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence

class sage.combinat.sloane_functions.A000272¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000290¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000292¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000302¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000312¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000326¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000330¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000396¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000578¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000583¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000587¶: Bases: sage.combinat.sloane_functions.ExponentialNumbers

class sage.combinat.sloane_functions.A000668¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000670¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000720¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A000796¶

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A000961¶

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A000984¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A001006¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A001045¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A001055¶

Bases: sage.combinat.sloane_functions.SloaneSequence

nwf(n, m)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A001109¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A001110¶

Bases: sage.combinat.sloane_functions.RecurrenceSequence

g(k)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A001147¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A001157¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A001189¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A001221¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A001222¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A001227¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A001333¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A001358¶

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A001405¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A001477¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A001694¶

Bases: sage.combinat.sloane_functions.SloaneSequence

is_powerful(n)¶

This function returns 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 Sloane’s OEIS A001694.

INPUT:

n - integer

OUTPUT:

True - if $n$ is a Powerful number, else False

EXAMPLES:

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

AUTHORS:

Jaap Spies (2006-12-07)

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A001836¶

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A001906¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A001909¶: Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence

class sage.combinat.sloane_functions.A001910¶: Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence

class sage.combinat.sloane_functions.A001969¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A002110¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A002113¶

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A002275¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A002378¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A002620¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A002808¶

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A003418¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A004086¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A004526¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A005100¶

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A005101¶

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A005117¶

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A005408¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A005843¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A006318¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A006530¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A006882¶

Bases: sage.combinat.sloane_functions.SloaneSequence

df()¶

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

EXAMPLES:

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

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A007318¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A008275¶

Bases: sage.combinat.sloane_functions.SloaneSequence

s(n, k)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A008277¶

Bases: sage.combinat.sloane_functions.SloaneSequence

s2(n, k)¶

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

EXAMPLES:

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

class sage.combinat.sloane_functions.A008683¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A010060¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A015521¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A015523¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A015530¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A015531¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A015551¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A018252¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A020639¶

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

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

Bases: sage.combinat.sloane_functions.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: a = sloane.A046660; a
Excess of n = Bigomega (with multiplicity) - omega (without multiplicity).
sage: a(0)
...
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]

AUTHORS:

Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A049310¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A051959¶

Bases: sage.combinat.sloane_functions.RecurrenceSequence

g(k)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A055790¶: Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2

class sage.combinat.sloane_functions.A061084¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A064553¶: Bases: sage.combinat.sloane_functions.SloaneSequence

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

Bases: sage.combinat.sloane_functions.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: 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]

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: a(-1)
...
ValueError: input n (=-1) must be a positive integer

AUTHORS:

Jaap Spies (2007-01-14)

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

Bases: sage.combinat.sloane_functions.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: 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]

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: a(0)
...
ValueError: input n (=0) must be a positive integer

AUTHORS:

Jaap Spies (2007-01-17)

class sage.combinat.sloane_functions.A082411¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A083103¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A083104¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A083105¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A083216¶: Bases: sage.combinat.sloane_functions.RecurrenceSequence2

class sage.combinat.sloane_functions.A090010¶: Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2

class sage.combinat.sloane_functions.A090012¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A090013¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A090014¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A090015¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A090016¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A111774¶

Bases: sage.combinat.sloane_functions.SloaneSequence

is_number_of_the_third_kind(n)¶

This function returns 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 False -

EXAMPLES:

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

AUTHORS:

Jaap Spies (2006-12-09)

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.A111775¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A111776¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.A111787¶: Bases: sage.combinat.sloane_functions.SloaneSequence

class sage.combinat.sloane_functions.ExponentialNumbers(a)¶: Bases: sage.combinat.sloane_functions.SloaneSequence

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

Bases: sage.combinat.sloane_functions.SloaneSequence

gen(a0, a1, d)¶

EXAMPLES:

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

list(n)¶

EXAMPLES:

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

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

Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence

gen(a0, a1, d)¶

EXAMPLES:

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

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

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

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

Bases: sage.combinat.sloane_functions.SloaneSequence

list(n)¶

EXAMPLES:

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

class sage.combinat.sloane_functions.Sloane¶

Bases: sage.structure.sage_object.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: len(sloane.trait_names()) > 100
True

And ensure none are being incorrectly returned:

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

Ensure we can access dynamic constructions and cache correctly:

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

And that we can access other functions in parent classes:

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

AUTHORS:

Nick Alexander

trait_names()¶

List Sloane generating functions for tab-completion. The member classes are inspected from module sage.combinat.sloane_functions.

They must be sub classes of SloaneSequence and must start with ‘A’. These restrictions are only to prevent typos, incorrect inspecting, etc.

EXAMPLES:

sage: type(sloane.trait_names())
<type 'list'>

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

Bases: sage.structure.sage_object.SageObject

Base class for a Sloane integer sequence.

EXAMPLES:

We create a dummy sequence:

list(n)¶

Return n terms of the sequence: sequence[offset], sequence[offset+1], ... , sequence[offset+n-1]. EXAMPLES:

sage: sloane.A000012.list(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 - non negative integer

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

EXAMPLES:

sage: from sage.combinat.sloane_functions import perm_mh
sage: perm_mh(3,3)
36
sage: perm_mh(3,4)
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: from sage.combinat.sloane_functions import recur_gen2
sage: it = recur_gen2(1,1,1,1)
sage: [it.next() for i in range(10)]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

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

inhomogenous 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: from sage.combinat.sloane_functions import recur_gen2b
sage: it = recur_gen2b(1,1,1,1, lambda n: 0)
sage: [it.next() for i in range(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: from sage.combinat.sloane_functions import recur_gen3
sage: it = recur_gen3(1,1,1,1,1,1)
sage: [it.next() for i in range(10)]
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]

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

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

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