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

%I A093998 #28 Jan 05 2025 19:51:37
%S A093998 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,
%T A093998 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,
%U A093998 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
%N A093998 Number of partitions of n with an even number of distinct Fibonacci parts.
%H A093998 Alois P. Heinz, <a href="/A093998/b093998.txt">Table of n, a(n) for n = 0..10946</a>
%H A093998 F. Ardila, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-3/quartardila03_2004.pdf">The coefficients of a Fibonacci power series</a>, Fib. Quart. 42 (3) (2004), 202-204.
%H A093998 N. Robbins, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/34-4/robbins.pdf">Fibonacci partitions</a>, Fib. Quart. 34 (4) (1996), 306-313.
%H A093998 J. Shallit, <a href="https://arxiv.org/abs/2007.14930">Robbins and Ardila meet Berstel</a>, Arxiv preprint arXiv:2007.14930 [math.CO], 2020.
%F A093998 G.f.: (Product_{k>=2} (1 + x^{F_k}) + Product_{k>=2} (1 - x^{F_k}))/2.
%p A093998 F:= combinat[fibonacci]:
%p A093998 b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<2, 0,
%p A093998        b(n, i-1, t)+`if`(F(i)>n, 0, b(n-F(i), i-1, 1-t))))
%p A093998     end:
%p A093998 a:= proc(n) local j; for j from ilog[(1+sqrt(5))/2](n+1)
%p A093998        while F(j+1)<=n do od; b(n, j, 1)
%p A093998     end:
%p A093998 seq(a(n), n=0..100);  # _Alois P. Heinz_, Jul 11 2013
%t A093998 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 *)
%Y A093998 Cf. A000119, A093997.
%K A093998 easy,nonn
%O A093998 0,12
%A A093998 _N. Sato_, May 24 2004
%E A093998 Edited and extended by _Robert G. Wilson v_, May 29 2004