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A164116
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Expansion of (1 - x) * (1 - x^4) / (1 - x^5) in powers of x.
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11
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1, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1
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OFFSET
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0,6
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COMMENTS
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This sequence with a(0) replaced by 2 appears, together with three other sequences, in the formula 2*exp(2*Pi*n*I/5) = 2*T(n,x) + S(n-1,x)*sqrt(2+phi)*I, with x = (phi-1)/2 and I = sqrt(-1), where phi = (1+sqrt(5))/2 (golden section) after reduction of powers using phi^2 = phi+1. T and S are Chebyshev polynomials from A053120 and A049310. This results in 2*exp(2*Pi*n*I/5) = (A(n) + B(n)*phi) + (C(n) + D(n)*phi)*sqrt(2+phi)*I, with A(n) = a(n+5), B(n) = A080891(n), C(n) = A156174(n+4) and D(n) = A010891(n+3) for n >= 0. - Wolfdieter Lang, Feb 26 2014
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LINKS
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FORMULA
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Euler transform of length 5 sequence [-1, 0, 0, -1, 1].
a(n) = a(-n) for all n in Z. a(n+5) = a(n) unless n=0 or n=-5.
G.f.: (1 - x^4)/(1 + x + x^2 + x^3 + x^4).
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EXAMPLE
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G.f. = 1 - x - x^4 + 2*x^5 - x^6 - x^9 + 2*x^10 - x^11 - x^14 + 2*x^15 - x^16 + ...
exp(2*Pi*3*I/5) = (0 - phi) + (1 - phi)*sqrt(2+phi)*I, with phi = (1+sqrt(5))/2. - Wolfdieter Lang, Feb 26 2014
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MATHEMATICA
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CoefficientList[Series[(1-x)(1-x^4)/(1-x^5), {x, 0, 110}], x] (* Harvey P. Dale, Sep 25 2013 *)
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PROG
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(PARI) {a(n) = -(n==0) + [2, -1, 0, 0, -1][n%5 + 1]};
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 - x^4)/(1+x+x^2+x^3+x^4))); // G. C. Greubel, Sep 22 2018
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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