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Total number of Fibonacci parts in all compositions of n.
2

%I #15 Mar 17 2024 08:05:30

%S 0,1,3,8,19,46,106,241,541,1198,2629,5724,12380,26625,56978,121413,

%T 257740,545308,1150272,2419856,5078336,10633921,22222338,46353669,

%U 96525324,200686620,416645184,863834256,1788756288,3699688128,7643727360,15776156928,32529718272

%N Total number of Fibonacci parts in all compositions of n.

%H Alois P. Heinz, <a href="/A309537/b309537.txt">Table of n, a(n) for n = 0..3312</a>

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

%F a(n) ~ c * 2^n * n, where c = 0.22756969930196647294851075611776578612085598114... - _Vaclav Kotesovec_, Aug 18 2019

%F c = A124091/4 - 3/8. - _Vaclav Kotesovec_, Mar 17 2024

%p a:= proc(n) option remember; add(a(n-j)+`if`((t->issqr(t+4)

%p or issqr(t-4))(5*j^2), ceil(2^(n-j-1)), 0), j=1..n)

%p end:

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

%t 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 a /@ Range[0, 33] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)

%Y Cf. A000045, A102291, A124091, A144115.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 06 2019