A093996 G.f.: Product_{k>=2} (1 - x^{F_k}) where F_k are the Fibonacci numbers.

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, -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, 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

Offset: 0

Author

N. Sato, May 24 2004

Keywords

• easy
• sign

Comments

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.

Every term is -1, 0 or 1.

Examples

			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
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• F. Ardila, The Coefficients of a Fibonacci power series, arXiv:math/0409418 [math.CO], 2004.
• Neville Robbins, Fibonacci partitions, The Fibonacci Quarterly, 34.4 (1996), pp. 306-313.
• Yufei Zhao, The coefficients of a truncated Fibonacci power series, Fib. Q., 46/47 (2008/2009), 53-55.

Crossrefs

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

Programs

Magma

R:=PowerSeriesRing(Integers(), 115); Coefficients(R!( (&*[1-x^Fibonacci(j): j in [2..13]]) )); // G. C. Greubel, Dec 27 2021

Mathematica

Take[ CoefficientList[ Expand[ Product[1 - x^Fibonacci[k], {k, 2, 13}]], x], 105] (* Robert G. Wilson v, May 29 2004 *)
nn = 11; Take[CoefficientList[Expand[Product[1 - x^Fibonacci[n], {n, 2, nn}]], x], Fibonacci[nn+1]] (* T. D. Noe, Feb 27 2014 *)

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

Formula

Ardila gives a fast recurrence.

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

Extensions

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