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

0,12

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.

Formula

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

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 *)

Crossrefs

Cf. A000119, A093997.

Sequence in context: A225089 A262436 A336499 * A247918 A237203 A339444

Adjacent sequences: A093995 A093996 A093997 * A093999 A094000 A094001

Keyword

easy,nonn

Author

N. Sato, May 24 2004

Extensions

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

Status

approved