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A334948
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a(n) is the number of partitions of n into consecutive parts that differ by 6.
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8
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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
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OFFSET
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1,8
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COMMENTS
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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.
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} x^(k*(3*k - 2)) / (1 - x^k). - Ilya Gutkovskiy, Nov 23 2020
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EXAMPLE
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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.
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MATHEMATICA
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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]];
Table[Count[IntegerPartitions[n], _?(Union[Abs[Differences[#]]]=={6}&)]+1, {n, 110}] (* Harvey P. Dale, Dec 07 2020 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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