A093996 - OEIS

%I #44 Jan 05 2025 19:51:37

%S 1,-1,-1,0,1,0,0,1,-1,0,0,1,-1,-1,1,0,0,0,1,-1,-1,0,1,1,-1,0,0,0,0,1,

%T -1,-1,0,1,0,0,1,0,-1,-1,1,0,0,0,0,0,0,1,-1,-1,0,1,0,0,1,-1,0,0,1,-1,

%U 0,0,-1,0,1,1,-1,0,0,0,0,0,0,0,0,0,1,-1,-1,0,1,0,0,1,-1,0,0,1,-1,-1,1,0,0,0,1,-1,-1,1,0,0,-1,1,0,0,1

%N G.f.: Product_{k>=2} (1 - x^{F_k}) where F_k are the Fibonacci numbers.

%C Number of partitions of n with an even number of distinct Fibonacci parts minus the number of partitions of n with an odd number of distinct Fibonacci parts.

%C Every term is -1, 0 or 1.

%H T. D. Noe, <a href="/A093996/b093996.txt">Table of n, a(n) for n = 0..1000</a>

%H F. Ardila, <a href="http://arxiv.org/abs/math/0409418">The Coefficients of a Fibonacci power series</a>, arXiv:math/0409418 [math.CO], 2004.

%H Neville Robbins, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/34-4/robbins.pdf">Fibonacci partitions</a>, The Fibonacci Quarterly, 34.4 (1996), pp. 306-313.

%H Yufei Zhao, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/46_47-1/Zhao_12-08.pdf">The coefficients of a truncated Fibonacci power series</a>, Fib. Q., 46/47 (2008/2009), 53-55.

%F Ardila gives a fast recurrence.

%F a(n) = A093998(n) - A093997(n).

%e G.f. = 1 - x - x^2 + x^4 + x^7 - x^8 + x^11 - x^12 - x^13 + x^14 + x^18 - x^19 - x^20 + x^22 + x^23 - x^24 + x^29 - x^30 - x^31 + x^33 + x^36 - x^38 - x^39 + x^40 + x^47 - ... - _N. J. A. Sloane_, May 30 2009

%t Take[ CoefficientList[ Expand[ Product[1 - x^Fibonacci[k], {k, 2, 13}]], x], 105] (* _Robert G. Wilson v_, May 29 2004 *)

%t nn = 11; Take[CoefficientList[Expand[Product[1 - x^Fibonacci[n], {n, 2, nn}]], x], Fibonacci[nn+1]] (* _T. D. Noe_, Feb 27 2014 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 115); Coefficients(R!( (&*[1-x^Fibonacci(j): j in [2..13]]) )); // _G. C. Greubel_, Dec 27 2021

%o (Sage) [( product( 1-x^fibonacci(j) for j in (2..14) ) ).series(x,n+1).list()[n] for n in (0..115)] # _G. C. Greubel_, Dec 27 2021

%Y Cf. A000045, A000119, A093997, A093998, A151661.

%K easy,sign

%O 0,1

%A _N. Sato_, May 24 2004

%E Edited and extended by _Robert G. Wilson v_, May 29 2004