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%I #14 Apr 04 2022 11:41:16
%S 1,1,1,2,4,10,43,713,79190,214727233,93022035568623,
%T 154947556313144561674052,151111389118856626519180244830493192189,
%U 317985304587719144905460952112822027368399484007468648195691719
%N Number of compositions of the n-th Fibonacci number into a Fibonacci number of Fibonacci parts.
%H Alois P. Heinz, <a href="/A344790/b344790.txt">Table of n, a(n) for n = 0..18</a>
%e a(4) = 4: [3], [2,1], [1,2], [1,1,1].
%e a(5) = 10: [5], [3,2], [3,1,1], [2,3], [2,2,1], [2,1,2], [1,3,1], [1,2,2], [1,1,3], [1,1,1,1,1].
%p f:= n-> (t-> issqr(t+4) or issqr(t-4))(5*n^2):
%p b:= proc(n, c) option remember; `if`(n=0, `if`(f(c),
%p 1, 0), add(`if`(f(j), b(n-j, c+1), 0), j=1..n))
%p end:
%p a:= n-> b((<<0|1>, <1|1>>^n)[1, 2], 0):
%p seq(a(n), n=0..13);
%t f[n_] := Function[t, IntegerQ@Sqrt[t+4] || IntegerQ@Sqrt[t-4]][5*n^2];
%t b[n_, c_] := b[n, c] = If[n == 0, If[f[c], 1, 0],
%t Sum[If[f[j], b[n-j, c+1], 0], {j, 1, n}]];
%t a[n_] := b[MatrixPower[{{0, 1}, {1, 1}}, n ][[1, 2]], 0];
%t Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Apr 04 2022, after _Alois P. Heinz_ *)
%Y Cf. A000045, A343537, A344782.
%K nonn
%O 0,4
%A _Alois P. Heinz_, May 28 2021
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