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

1,8

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.

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

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.

Sequence in context: A346094 A082898 A095999 * A161238 A161062 A161266

Adjacent sequences: A334945 A334946 A334947 * A334949 A334950 A334951

Keyword

nonn

Author

Omar E. Pol, May 27 2020

Status

approved