A334732 a(n) is the total number of parts in all partitions of n into consecutive parts that differ by 5.

1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 4, 3, 1, 6, 1, 3, 4, 3, 1, 6, 1, 3, 4, 3, 1, 6, 5, 3, 4, 3, 5, 6, 1, 3, 8, 3, 1, 6, 5, 3, 4, 3, 5, 6, 1, 3, 8, 8, 1, 6, 5, 3, 9, 3, 5, 6, 1, 8, 8, 3, 1, 6, 10, 3, 4, 3, 5, 11, 1, 3, 8, 3, 6, 12, 5, 3, 4, 8, 5, 12, 1, 3, 13, 3, 1, 12

Offset: 1

Author

Omar E. Pol, May 09 2020

Keywords

• nonn

Comments

The one-part partition n = n is included in the count.

For the relation to heptagonal numbers see also A334540.

Examples

			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]. The number of parts of these partitions are 1, 2, 3 respectively and the total number of parts is 1 + 2 + 3 = 6, so the a(27) = 6.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Row sums of A334540.

Column k=5 of A334466.

Sequences of the same family whose consecutive parts differs by k are: A000203 (k=0), A204217 (k=1), A066839 (k=2), A330889 (k=3), A334464 (k=4), this sequence (k=5), A334949 (k=6).

Cf. A000566, A334465, A334540, A334541, A334733.

Programs

Maple

A334732 := proc(n)
    local a,k;
    a := 0 ;
    for k from 1 do
        if n>= A000566(k) then
            a := a+A334540(n,k);
        else
            return a;
        end if;
    end do:
end proc:
seq(A334732(n),n=1..120) ; # R. J. Mathar, Oct 02 2020

Mathematica

nmax = 100;
CoefficientList[Sum[n x^(n(5n-3)/2-1)/(1-x^n), {n, 1, nmax}]+O[x]^nmax, x] (* Jean-François Alcover, Nov 30 2020 *)
Table[Sum[If[n > 5*k*(k-1)/2 && IntegerQ[n/k - 5*(k-1)/2], k, 0], {k, Divisors[2*n]}], {n, 1, 100}] (* Vaclav Kotesovec, Oct 23 2024 *)

PARI

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, k*x^(k*(5*k-3)/2)/(1-x^k))) \\ Seiichi Manyama, Dec 04 2020

Formula

G.f.: Sum_{n>=1} n*x^(n*(5*n-3)/2)/(1-x^n) (for proof see A330889).

Sum_{k=1..n} a(k) ~ 2^(3/2) * n^(3/2) / (3*sqrt(5)). - Vaclav Kotesovec, Oct 23 2024

Extensions

More terms from R. J. Mathar, Oct 02 2020