A319503 Number of partitions of Fibonacci(n) into exactly n positive Fibonacci numbers.

1, 1, 0, 0, 0, 1, 2, 6, 16, 43, 117, 305, 769, 1907, 4686, 11587, 28580, 70451, 172880, 423629, 1036332, 2533559, 6186635, 15092985, 36784586, 89590410, 218069921, 530551804, 1290218120, 3136385254, 7621522229, 18515039477, 44966884766, 109184448962

Offset

0,7

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

a(n) = [x^A000045(n) y^n] 1/Product_{j>=2} (1-y*x^A000045(j)).

a(n) = A319394(A000045(n),n).

Example

a(0) = 1: the empty partition.
a(1) = 1: 1.
a(5) = 1: 11111.
a(6) = 2: 221111, 311111.
a(7) = 6: 2222221, 3222211, 3322111, 3331111, 5221111, 5311111.

Mathematica

(* Program not suitable for a large number of terms. *)
a[n_] := a[n] = If[n < 2, 1, IntegerPartitions[Fibonacci[n], {n}, Fibonacci[Range[2, n - 1]]] //Length];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 24}] (* Jean-François Alcover, Dec 08 2020 *)

Crossrefs

Cf. A000045, A098641, A259254, A319394, A319435.

Sequence in context: A244399 A291142 A027994 * A295572 A372191 A027068

Adjacent sequences: A319500 A319501 A319502 * A319504 A319505 A319506

Keyword

nonn

Author

Alois P. Heinz, Sep 20 2018

Status

approved