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A188510
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Expansion of x*(1 + x^2)/(1 + x^4) in powers of x.
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17
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0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0
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OFFSET
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0,1
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LINKS
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FORMULA
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Euler transform of length 8 sequence [0, 1, 0, -2, 0, 0, 0, 1].
a(n) is multiplicative with a(2^e) = 0^e, a(p^e) = 1 if p == 1 or 3 (mod 8), a(p^e) = (-1)^e if p == 5 or 7 (mod 8).
G.f.: x * (1 - x^4)^2/((1 - x^2)*(1 - x^8)) = (x + x^3)/(1 + x^4).
a(-n) = -a(n) = a(n+4).
G.f.: x/(1 - x^2/(1 + 2*x^2/(1 - x^2))). - Michael Somos, Jan 03 2013
a(n) = ((-2)/n), where (k/n) is the Kronecker symbol. Period 8. See the Eric Weisstein link. - Wolfdieter Lang, May 29 2013
a(n) = sqrt(2)*sin(Pi*n/2)*cos(Pi*n/4).
E.g.f.: sqrt(2)*sin(x/sqrt(2))*cosh(x/sqrt(2)).
a(n) = ((-2)^(2*i+1)/n) for all integers i >= 0, where (k/n) is the Kronecker symbol. (End)
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EXAMPLE
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G.f. = x + x^3 - x^5 - x^7 + x^9 + x^11 - x^13 - x^15 + x^17 + x^19 - x^21 + ...
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MATHEMATICA
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Table[KroneckerSymbol[-2, n], {n, 0, 104}] (* Wolfdieter Lang, May 30 2013 *)
a[ n_] := Mod[n, 2] (-1)^Quotient[ n, 4]; (* Michael Somos, Apr 17 2015 *)
CoefficientList[Series[x*(1+x^2)/(1+x^4), {x, 0, 60}], x] (* G. C. Greubel, Aug 02 2018 *)
LinearRecurrence[{0, 0, 0, -1}, {0, 1, 0, 1}, 120] (* or *) PadRight[{}, 120, {0, 1, 0, 1, 0, -1, 0, -1}] (* Harvey P. Dale, Jan 25 2023 *)
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PROG
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(PARI) {a(n) = (n%2) * (-1)^(n\4)};
(PARI) x='x+O('x^60); concat([0], Vec(x*(1+x^2)/(1+x^4))) \\ G. C. Greubel, Aug 02 2018
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x^2)/(1+x^4))); // G. C. Greubel, Aug 02 2018
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CROSSREFS
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KEYWORD
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sign,easy,mult
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AUTHOR
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STATUS
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approved
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