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

0,3

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

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

Sequence in context: A054127 A184052 A295057 * A018012 A359673 A378829

Adjacent sequences: A309532 A309533 A309534 * A309536 A309537 A309538

Keyword

nonn

Author

Alois P. Heinz, Aug 06 2019

Status

approved