A341072 - OEIS

%I #13 May 02 2022 03:03:36

%S 1,1,3,7,23,71,231,750,2479,8251,27673,93248,315515,1071097,3646618,

%T 12445982,42571327,145895599,500855361,1722062265,5929045173,

%U 20439121983,70539320558,243695962031,842704577995,2916613479471,10102511916071,35018749192885

%N Number of compositions of 2n into n Fibonacci parts.

%H Alois P. Heinz, <a href="/A341072/b341072.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A121548(2n,n).

%F a(n) ~ c * d^n / sqrt(n), where d = 3.532272846853808150678856189005437981671101510837727... and c = 0.2903295565097076269212760734928134309226027... - _Vaclav Kotesovec_, Feb 14 2021

%p g:= proc(n) g(n):= (t-> issqr(t+4) or issqr(t-4))(5*n^2) end:

%p b:= proc(n, t) option remember;

%p       `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(

%p       `if`(g(j), b(n-j, t-1), 0), j=1..n)))

%p     end:

%p a:= n-> b(2*n, n):

%p seq(a(n), n=0..35);

%t g[n_] := g[n] = With[{t = 5*n^2}, IntegerQ@Sqrt[t+4] || IntegerQ@Sqrt[t-4]];

%t b[n_, t_] := b[n, t] =

%t      If[n == 0, If[t == 0, 1, 0], If[t < 1, 0, Sum[

%t      If[g[j], b[n - j, t - 1], 0], {j, 1, n}]]];

%t a[n_] := b[2n, n];

%t Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, May 02 2022, after _Alois P. Heinz_ *)

%Y Cf. A000045, A121548, A341071.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Feb 04 2021