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

0, 1, 3, 8, 19, 46, 106, 241, 541, 1198, 2629, 5724, 12380, 26625, 56978, 121413, 257740, 545308, 1150272, 2419856, 5078336, 10633921, 22222338, 46353669, 96525324, 200686620, 416645184, 863834256, 1788756288, 3699688128, 7643727360, 15776156928, 32529718272

Offset: 0

Author

Alois P. Heinz, Aug 06 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3312

Crossrefs

Cf. A000045, A102291, A124091, A144115.

Programs

Maple

a:= proc(n) option remember; add(a(n-j)+`if`((t->issqr(t+4)
      or issqr(t-4))(5*j^2), ceil(2^(n-j-1)), 0), j=1..n)
    end:
seq(a(n), n=0..33);

Mathematica

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}];
a /@ Range[0, 33] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

Formula

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

a(n) ~ c * 2^n * n, where c = 0.22756969930196647294851075611776578612085598114... - Vaclav Kotesovec, Aug 18 2019

c = A124091/4 - 3/8. - Vaclav Kotesovec, Mar 17 2024