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A309535

Total number of square parts in all compositions of n.

0, 1, 2, 5, 13, 30, 69, 156, 348, 769, 1682, 3653, 7884, 16924, 36160, 76944, 163137, 344770, 726533, 1527052, 3202076, 6700096, 13992080, 29167936, 60703424, 126141953, 261754114, 542448645, 1122778124, 2321317916, 4794159168, 9891365008, 20388823360

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OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..3313

FORMULA

G.f.: Sum_{k>=1} x^(k^2)*(1-x)^2/(1-2*x)^2.

a(n) ~ c * 2^n * n, where c = (EllipticTheta[3, 0, 1/2] - 1)/8 = 0.1411171034014846448336823185681189155765645674... - Vaclav Kotesovec, Aug 18 2019, updated Mar 17 2024

a(n) = Sum_{k=1..A000196(n)} A045623(n-k^2). - Gregory L. Simay, Jun 07 2021

EXAMPLE

a(4) = 13: (1)(1)(1)(1), (1)(1)2, (1)2(1), 2(1)(1), 22, (1)3, 3(1), (4).

MAPLE

a:= proc(n) option remember; add(a(n-j)+

`if`(issqr(j), ceil(2^(n-j-1)), 0), j=1..n)

end:

seq(a(n), n=0..33);

MATHEMATICA

CoefficientList[Series[(EllipticTheta[3, 0, x]-1)*(1-x)^2/(2*(1-2*x)^2), {x, 0, 30}], x] (* Vaclav Kotesovec, Aug 18 2019 *)

Table[Sum[If[k == n, 1, (2^(n - k - 2)*(3 + n - k))] * If[IntegerQ[Sqrt[k]], 1, 0], {k, 1, n}], {n, 0, 30}] (* Vaclav Kotesovec, Aug 18 2019 *)

CROSSREFS