

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*kd+2)/2) / (1x^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 [27], [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[105] (* JeanFrançois 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*kd+2)/2)/(1x^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



