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 A334541 a(n) is the number of partitions of n into consecutive parts that differ by 5. 8
 1, 1, 1, 1, 1, 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, 2, 2, 2, 2, 2, 3, 1, 2, 3, 2, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 3, 3, 1, 3, 2, 2, 3, 2, 2, 3, 1, 3, 3, 2, 1, 3, 3, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2, 4, 2, 2, 2, 3, 2, 4, 1, 2, 4, 2, 1, 4, 2, 3, 2, 2, 2, 4, 2, 2, 3, 2, 1, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS Note that all sequences of this family as A000005, A001227, A038548, A117277, A334461, 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 heptagonal numbers see also A334465. LINKS David A. Corneth, 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 = 27 there are three partitions of 27 into consecutive parts that differ by 5, including 27 as a valid partition. They are , [16, 11] and [14, 9, 4], so a(27) = 3. MATHEMATICA first[n_] := Module[{res = Array[1&, n]}, For[i = 2, True, i++, start = i + 5 Binomial[i, 2]; If[start > n, Return[res]]; For[j = start, j <= n, j += i, res[[j]]++]]]; first (* Jean-François Alcover, Nov 30 2020, after David A. Corneth *) 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, 5) \\ Joerg Arndt, May 06 2020 (PARI) first(n) = { my(res = vector(n, i, 1)); for(i = 2, oo, start = i + 5 * binomial(i, 2); if(start > n, return(res)); forstep(j = start, n, i, res[j]++ ) ); } \\ David A. Corneth, May 17 2020 CROSSREFS Row sums of A334465. Column k=5 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), this sequence (k=5). Cf. A000041, A000566, A303300. Sequence in context: A161303 A161278 A160982 * A175150 A161236 A161060 Adjacent sequences:  A334538 A334539 A334540 * A334542 A334543 A334544 KEYWORD nonn,easy AUTHOR Omar E. Pol, May 05 2020 STATUS approved

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Last modified November 29 18:41 EST 2021. Contains 349416 sequences. (Running on oeis4.)