A344790 Number of compositions of the n-th Fibonacci number into a Fibonacci number of Fibonacci parts.

1, 1, 1, 2, 4, 10, 43, 713, 79190, 214727233, 93022035568623, 154947556313144561674052, 151111389118856626519180244830493192189, 317985304587719144905460952112822027368399484007468648195691719

Offset: 0

Author

Alois P. Heinz, May 28 2021

Keywords

• nonn

Examples

			a(4) = 4: [3], [2,1], [1,2], [1,1,1].
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].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..18

Crossrefs

Cf. A000045, A343537, A344782.

Programs

Maple

f:= n-> (t-> issqr(t+4) or issqr(t-4))(5*n^2):
b:= proc(n, c) option remember; `if`(n=0, `if`(f(c),
      1, 0), add(`if`(f(j), b(n-j, c+1), 0), j=1..n))
    end:
a:= n-> b((<<0|1>, <1|1>>^n)[1, 2], 0):
seq(a(n), n=0..13);

Mathematica

f[n_] := Function[t, IntegerQ@Sqrt[t+4] || IntegerQ@Sqrt[t-4]][5*n^2];
b[n_, c_] := b[n, c] = If[n == 0, If[f[c], 1, 0],
     Sum[If[f[j], b[n-j, c+1], 0], {j, 1, n}]];
a[n_] := b[MatrixPower[{{0, 1}, {1, 1}}, n ][[1, 2]], 0];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Apr 04 2022, after Alois P. Heinz *)