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A309537 Total number of Fibonacci parts in all compositions of n.

%I A309537 #15 Mar 17 2024 08:05:30
%S A309537 0,1,3,8,19,46,106,241,541,1198,2629,5724,12380,26625,56978,121413,
%T A309537 257740,545308,1150272,2419856,5078336,10633921,22222338,46353669,
%U A309537 96525324,200686620,416645184,863834256,1788756288,3699688128,7643727360,15776156928,32529718272
%N A309537 Total number of Fibonacci parts in all compositions of n.
%H A309537 Alois P. Heinz, <a href="/A309537/b309537.txt">Table of n, a(n) for n = 0..3312</a>
%F A309537 G.f.: Sum_{k>=2} x^Fibonacci(k)*(1-x)^2/(1-2*x)^2.
%F A309537 a(n) ~ c * 2^n * n, where c = 0.22756969930196647294851075611776578612085598114... - _Vaclav Kotesovec_, Aug 18 2019
%F A309537 c = A124091/4 - 3/8. - _Vaclav Kotesovec_, Mar 17 2024
%p A309537 a:= proc(n) option remember; add(a(n-j)+`if`((t->issqr(t+4)
%p A309537       or issqr(t-4))(5*j^2), ceil(2^(n-j-1)), 0), j=1..n)
%p A309537     end:
%p A309537 seq(a(n), n=0..33);
%t A309537 a[n_] := a[n] = Sum[a[n - j] + With[{t = 5 j^2}, If[IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4], Ceiling[2^(n - j - 1)], 0]], {j, 1, n}];
%t A309537 a /@ Range[0, 33] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)
%Y A309537 Cf. A000045, A102291, A124091, A144115.
%K A309537 nonn
%O A309537 0,3
%A A309537 _Alois P. Heinz_, Aug 06 2019