A093998 Number of partitions of n with an even number of distinct Fibonacci parts.

1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 1, 2, 1, 0, 2, 2, 2, 2, 1, 2, 2, 1, 3, 1, 1, 2, 1, 2, 2, 2, 3, 1, 2, 3, 1, 3, 2, 2, 3, 1, 3, 2, 1, 3, 2, 2, 2, 1, 2, 2, 2, 4, 1, 3, 3, 1, 4, 3, 3, 3, 1, 3, 3, 2, 4, 2, 3, 3, 1, 4, 2, 2, 4, 2, 3, 3, 2, 3, 2, 2, 3, 0, 2, 3, 2, 4, 2, 4, 3, 1, 5, 3, 3, 4, 2, 4, 4, 3

Offset: 0

Author

N. Sato, May 24 2004

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10946
• F. Ardila, The coefficients of a Fibonacci power series, Fib. Quart. 42 (3) (2004), 202-204.
• N. Robbins, Fibonacci partitions, Fib. Quart. 34 (4) (1996), 306-313.
• J. Shallit, Robbins and Ardila meet Berstel, Arxiv preprint arXiv:2007.14930 [math.CO], 2020.

Crossrefs

Cf. A000119, A093997.

Programs

Maple

F:= combinat[fibonacci]:
b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<2, 0,
       b(n, i-1, t)+`if`(F(i)>n, 0, b(n-F(i), i-1, 1-t))))
    end:
a:= proc(n) local j; for j from ilog[(1+sqrt(5))/2](n+1)
       while F(j+1)<=n do od; b(n, j, 1)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 11 2013

Mathematica

Take[ CoefficientList[ Expand[ Product[1 + x^Fibonacci[k], {k, 2, 13}]/2 + Product[1 - x^Fibonacci[k], {k, 2, 13}]/2], x], 105] (* Robert G. Wilson v, May 29 2004 *)

Formula

G.f.: (Product_{k>=2} (1 + x^{F_k}) + Product_{k>=2} (1 - x^{F_k}))/2.

Extensions

Edited and extended by Robert G. Wilson v, May 29 2004