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A309535 Total number of square parts in all compositions of n.

%I A309535 #28 Mar 17 2024 07:31:05
%S A309535 0,1,2,5,13,30,69,156,348,769,1682,3653,7884,16924,36160,76944,163137,
%T A309535 344770,726533,1527052,3202076,6700096,13992080,29167936,60703424,
%U A309535 126141953,261754114,542448645,1122778124,2321317916,4794159168,9891365008,20388823360
%N A309535 Total number of square parts in all compositions of n.
%H A309535 Alois P. Heinz, <a href="/A309535/b309535.txt">Table of n, a(n) for n = 0..3313</a>
%F A309535 G.f.: Sum_{k>=1} x^(k^2)*(1-x)^2/(1-2*x)^2.
%F A309535 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
%F A309535 a(n) = Sum_{k=1..A000196(n)} A045623(n-k^2). - _Gregory L. Simay_, Jun 07 2021
%e A309535 a(4) = 13: (1)(1)(1)(1), (1)(1)2, (1)2(1), 2(1)(1), 22, (1)3, 3(1), (4).
%p A309535 a:= proc(n) option remember; add(a(n-j)+
%p A309535       `if`(issqr(j), ceil(2^(n-j-1)), 0), j=1..n)
%p A309535     end:
%p A309535 seq(a(n), n=0..33);
%t A309535 CoefficientList[Series[(EllipticTheta[3, 0, x]-1)*(1-x)^2/(2*(1-2*x)^2), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Aug 18 2019 *)
%t A309535 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 *)
%Y A309535 Cf. A000196, A000290, A045623, A073336, A102291.
%K A309535 nonn
%O A309535 0,3
%A A309535 _Alois P. Heinz_, Aug 06 2019