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

%I A334948 #34 Oct 22 2024 12:55:27
%S A334948 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,
%T A334948 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,
%U A334948 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
%N A334948 a(n) is the number of partitions of n into consecutive parts that differ by 6.
%C A334948 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.
%C A334948 For the relation to octagonal numbers see also A334946.
%H A334948 Seiichi Manyama, <a href="/A334948/b334948.txt">Table of n, a(n) for n = 1..10000</a>
%F A334948 G.f.: Sum_{k>=1} x^(k*(3*k - 2)) / (1 - x^k). - _Ilya Gutkovskiy_, Nov 23 2020
%e A334948 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.
%t A334948 nmax = 105;
%t A334948 col[k_] := col[k] = CoefficientList[Sum[x^(n(k n - k + 2)/2 - 1)/(1 - x^n), {n, 1, nmax}] + O[x]^nmax, x];
%t A334948 a[n_] := col[6][[n]];
%t A334948 Array[a, nmax] (* _Jean-François Alcover_, Nov 30 2020 *)
%t A334948 Table[Count[IntegerPartitions[n],_?(Union[Abs[Differences[#]]]=={6}&)]+1,{n,110}] (* _Harvey P. Dale_, Dec 07 2020 *)
%t A334948 Table[Sum[If[n > 3*k*(k-1), 1, 0], {k, Divisors[n]}], {n, 1, 100}] (* _Vaclav Kotesovec_, Oct 22 2024 *)
%o A334948 (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
%Y A334948 Row sums of A334946.
%Y A334948 Column k=6 of A323345.
%Y A334948 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).
%Y A334948 Cf. A000041, A000567, A303300, A334947,  A334949, A334953.
%K A334948 nonn
%O A334948 1,8
%A A334948 _Omar E. Pol_, May 27 2020