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A093996
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G.f.: Product_{k>=2} (1 - x^{F_k}) where F_k are the Fibonacci numbers.
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7
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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Ardila gives a fast recurrence.
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(Magma) R<x>:=PowerSeriesRing(Integers(), 115); Coefficients(R!( (&*[1-x^Fibonacci(j): j in [2..13]]) )); // G. C. Greubel, Dec 27 2021
(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
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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