A338731 -id:A338731 - cp's OEIS frontend

A117277 Number of partitions of n whose consecutive parts differ by 3.

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 3, 2, 1, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 1, 4, 2, 2, 2, 2, 3, 4, 1, 2, 3, 3, 1, 4, 2, 2, 3, 2, 2, 4, 1, 3, 3, 2, 1, 4, 4, 2, 2, 2, 2, 5, 1, 3, 3, 2, 2, 4, 2, 2, 3, 3, 2, 4, 1, 2, 4, 3, 2, 4, 2, 3, 2, 2, 3, 4, 3, 2, 3, 2, 1, 6

Offset: 1

Emeric Deutsch, Mar 07 2006

nonn

Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly 3 times. Example: a(15)=3 because we have [3,3,2,2,2,1,1,1],[2,2,2,2,2,2,1,1,1] and [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].
Row sums of A330887. - Omar E. Pol, May 07 2020
Column 3 of A323345. - Omar E. Pol, Dec 03 2020

			a(15) = 3 because we have [15], [9,6] and [8,5,2].

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A001227, A038548, A323345, A330887, A338731.

g:=sum(x^((3*k^2-k)/2)/(1-x^k),k=1..10): gser:=series(g,x=0,140): seq(coeff(gser,x^n),n=1..135);

Table[Sum[If[n > 3*k*(k-1)/2 && IntegerQ[n/k - 3*(k-1)/2], 1, 0], {k, Divisors[2*n]}], {n, 1, 100}] (* Vaclav Kotesovec, Oct 23 2024 *)

seq(N,d)=my(x='x+O('x^N));Vec(sum(k=1,N,x^(k*(d*k-d+2)/2)/(1-x^k)));
seq(100,3) \\ Joerg Arndt, May 05 2020

G.f.: Sum_{k>=1} x^((3*k^2-k)/2)/(1-x^k). In general, the generating function for the number of partitions in which consecutive parts differ by d is Sum_{k>=1} x^(k*(d*k-d+2)/2)/(1-x^k). For d=0, 1 and 2 one obtains A000005, A001227 and A038548, respectively.

A338730 Generating function Sum_{n >= 1} a(n)x^n = Sum_{k>=1} kx^(k(3k+1)/2)/(1-x^k).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 4, 3, 1, 6, 1, 3, 4, 3, 5, 6, 1, 3, 8, 3, 1, 6, 5, 3, 4, 3, 5, 6, 6, 3, 8, 3, 1, 11, 5, 3, 4, 3, 10, 6, 1, 3, 8, 8, 1, 12, 5, 3, 9, 3, 5, 12, 1, 8, 8, 3, 1, 12, 10, 3, 4, 3, 5, 17, 1, 10, 8, 3, 6, 12, 5, 3, 11, 8, 5, 12, 1, 3

Offset: 0

N. J. A. Sloane, Dec 02 2020

nonn

The OEIS contains many very similar sequences, but this one was missing.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A001227, A081757, A117277, A330889, A338731, A338732.

my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, k*x^(k*(3*k+1)/2)/(1-x^k)))) \\ Seiichi Manyama, Dec 03 2020

A338732 Generating function Sum_{n >= 0} a(n)x^n = Sum_{k>=1} x^(k(3*k+1))/(1-x^k).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 4, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 1, 3

Offset: 0

N. J. A. Sloane, Dec 03 2020

nonn

The OEIS contains many very similar sequences, but this one was missing.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A001227, A081757, A117277, A330889, A338730, A338731.

my(N=66, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(k*(3*k+1))/(1-x^k)))) \\ Seiichi Manyama, Dec 04 2020

cp's OEIS Frontend

A117277 Number of partitions of n whose consecutive parts differ by 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A338730 Generating function Sum_{n >= 1} a(n)x^n = Sum_{k>=1} kx^(k(3k+1)/2)/(1-x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A338732 Generating function Sum_{n >= 0} a(n)x^n = Sum_{k>=1} x^(k(3*k+1))/(1-x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

cp's OEIS Frontend

A117277 Number of partitions of n whose consecutive parts differ by 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A338730 Generating function Sum_{n >= 1} a(n)*x^n = Sum_{k>=1} k*x^(k*(3*k+1)/2)/(1-x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A338732 Generating function Sum_{n >= 0} a(n)*x^n = Sum_{k>=1} x^(k*(3*k+1))/(1-x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A338730 Generating function Sum_{n >= 1} a(n)x^n = Sum_{k>=1} kx^(k(3k+1)/2)/(1-x^k).

A338732 Generating function Sum_{n >= 0} a(n)x^n = Sum_{k>=1} x^(k(3*k+1))/(1-x^k).