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

0,3

Links

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

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

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 *)

Crossrefs

Cf. A000045, A102291, A124091, A144115.

Sequence in context: A026789 A308372 A096576 * A126874 A284942 A121811

Adjacent sequences: A309534 A309535 A309536 * A309538 A309539 A309540

Keyword

nonn

Author

Alois P. Heinz, Aug 06 2019

Status

approved