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 A334461 a(n) is the number of partitions of n into consecutive parts that differ by 4. 10
 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 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, 3, 2, 1, 4, 1, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 5, 1, 2, 2, 3, 2, 4, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 1, 4, 1, 4, 2, 2, 1, 5, 2, 2, 2, 3, 1, 5, 2, 3, 2, 2, 2, 5, 1, 3, 2, 4, 1, 4, 1, 3, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 LINKS Seiichi Manyama, Table of n, a(n) for n = 1..10000 FORMULA The g.f. for "consecutive parts that differ by d" is Sum_{k>=1} x^(k*(d*k-d+2)/2) / (1-x^k); cf. A117277. - Joerg Arndt, Nov 30 2020 EXAMPLE For n = 28 there are three partitions of 28 into consecutive parts that differ by 4, including 28 as a valid partition. They are [28], [16, 12] and [13, 9, 5, 1]. So a(28) = 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[4][[n]]; Array[a, nmax] (* Jean-François Alcover, Nov 30 2020 *) PROG (PARI) 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, 4) \\ Joerg Arndt, May 05 2020 CROSSREFS Row sums of A334460. Column k=4 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), this sequence (k=4), A334541 (k=5), A334948 (k=6). Sequence in context: A086435 A266226 A099305 * A338652 A033109 A321469 Adjacent sequences: A334458 A334459 A334460 * A334462 A334463 A334464 KEYWORD nonn AUTHOR Omar E. Pol, May 01 2020 STATUS approved

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Last modified July 15 00:37 EDT 2024. Contains 374323 sequences. (Running on oeis4.)