A359516 Number of compositions (ordered partitions) of n into at most 4 positive Fibonacci numbers (with a single type of 1).

1, 1, 2, 4, 7, 13, 20, 27, 35, 40, 46, 50, 55, 60, 61, 60, 65, 68, 72, 76, 73, 72, 66, 66, 79, 73, 85, 80, 79, 90, 76, 84, 85, 60, 72, 56, 69, 85, 69, 99, 89, 70, 97, 73, 94, 97, 66, 90, 72, 70, 96, 60, 85, 60, 24, 72, 44, 71, 88, 57, 105, 85, 78, 111, 74, 97, 82, 48, 97, 69, 79

Offset: 0

Author

Ilya Gutkovskiy, Jan 03 2023

Keywords

• nonn

Crossrefs

Cf. A000045, A076739, A121548, A357688, A359513, A359514, A359515.

Programs

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, 1, `if`(t<1, 0,
      add(`if`(g(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 03 2023

Mathematica

g[n_] := Function[t, IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4]][5 n^2];
b[n_, t_] := b[n, t] = If[n == 0, 1, If[t < 1, 0, Sum[If[g[j], b[n - j, t - 1], 0], {j, 1, n}]]];
a[n_] := b[n, 4];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 13 2023, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..4} A121548(n,k). - Alois P. Heinz, Jan 03 2023