A334948 a(n) is the number of partitions of n into consecutive parts that differ by 6.

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

Offset: 1

Author

Omar E. Pol, May 27 2020

Keywords

• nonn

Comments

Note that all sequences of this family as A000005, A001227, A038548, A117277, A334461, A334541, etc. could be prepended with a(0) = 1 when they are interpreted as sequences of number of partitions, since A000041(0) = 1. However here a(0) is omitted in accordance with the mentioned members of the same family.

For the relation to octagonal numbers see also A334946.

Examples

			For n = 24 there are three partitions of 24 into consecutive parts that differ by 6, including 24 as a valid partition. They are [24], [15, 9] and [14, 8, 2], so a(24) = 3.
		

Links

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

Crossrefs

Row sums of A334946.

Column k=6 of A323345.

Sequences of this family whose consecutive parts differ by k are A000005 (k=0), A001227 (k=1), A038548 (k=2), A117277 (k=3), A334461 (k=4), A334541 (k=5), this sequence (k=6).

Cf. A000041, A000567, A303300, A334947, A334949, A334953.

Programs

Mathematica

nmax = 105;
col[k_] := col[k] = CoefficientList[Sum[x^(n(k n - k + 2)/2 - 1)/(1 - x^n), {n, 1, nmax}] + O[x]^nmax, x];
a[n_] := col[6][[n]];
Array[a, nmax] (* Jean-François Alcover, Nov 30 2020 *)
Table[Count[IntegerPartitions[n],?(Union[Abs[Differences[#]]]=={6}&)]+1,{n,110}] (* _Harvey P. Dale, Dec 07 2020 *)
Table[Sum[If[n > 3*k*(k-1), 1, 0], {k, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Oct 22 2024 *)

PARI

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^(k*(3*k-2))/(1-x^k))) \\ Seiichi Manyama, Dec 04 2020

Formula

G.f.: Sum_{k>=1} x^(k*(3*k - 2)) / (1 - x^k). - Ilya Gutkovskiy, Nov 23 2020