login
A093998
Number of partitions of n with an even number of distinct Fibonacci parts.
4
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
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
Sequence in context: A225089 A262436 A336499 * A247918 A237203 A339444
KEYWORD
easy,nonn
AUTHOR
N. Sato, May 24 2004
EXTENSIONS
Edited and extended by Robert G. Wilson v, May 29 2004
STATUS
approved