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

Offset: 0

Author

Alois P. Heinz, Aug 06 2019

Keywords

• nonn

Examples

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

Links

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

Crossrefs

Cf. A000196, A000290, A045623, A073336, A102291.

Programs

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 *)

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