A341072 Number of compositions of 2n into n Fibonacci parts.
1, 1, 3, 7, 23, 71, 231, 750, 2479, 8251, 27673, 93248, 315515, 1071097, 3646618, 12445982, 42571327, 145895599, 500855361, 1722062265, 5929045173, 20439121983, 70539320558, 243695962031, 842704577995, 2916613479471, 10102511916071, 35018749192885
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
Programs
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Maple
g:= proc(n) g(n):= (t-> issqr(t+4) or issqr(t-4))(5*n^2) end: b:= proc(n, t) option remember; `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add( `if`(g(j), b(n-j, t-1), 0), j=1..n))) end: a:= n-> b(2*n, n): seq(a(n), n=0..35); -
Mathematica
g[n_] := g[n] = With[{t = 5*n^2}, IntegerQ@Sqrt[t+4] || IntegerQ@Sqrt[t-4]]; b[n_, t_] := b[n, t] = If[n == 0, If[t == 0, 1, 0], If[t < 1, 0, Sum[ If[g[j], b[n - j, t - 1], 0], {j, 1, n}]]]; a[n_] := b[2n, n]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)
Formula
a(n) = A121548(2n,n).
a(n) ~ c * d^n / sqrt(n), where d = 3.532272846853808150678856189005437981671101510837727... and c = 0.2903295565097076269212760734928134309226027... - Vaclav Kotesovec, Feb 14 2021