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A084099
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Expansion of (1+x)^2/(1+x^2).
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8
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1, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0
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OFFSET
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0,2
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COMMENTS
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Transform of sqrt(1+2x)/sqrt(1-2x) (A063886) under the Chebyshev transformation A(x)->((1-x^2)/(1+x^2))*A(x/(1+x^2)). - Paul Barry, Oct 12 2004
Euler transform of length 4 sequence [2, -3, 0, 1]. - Michael Somos, Aug 04 2009
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LINKS
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FORMULA
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G.f.: (1+x)^2/(1+x^2).
G.f.: 4*x + 2/(1+x)/G(0), where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 19 2013
a(n) = (1-sign(n)*(-1)^n)*(-1)^floor(n/2).
a(n) = 2*(n mod 2)*(-1)^floor(n/2) for n>0, a(0)=1.
a(n) = (1-(-1)^n)*(-1)^(n*(n-1)/2) for n>0, a(0)=1. (End)
a(n) = -a(n-2).
a(n) = i*((-i)^n-i^n) for n>0, where i = sqrt(-1).
(End)
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EXAMPLE
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G.f. = 1 + 2*x - 2*x^3 + 2*x^5 - 2*x^7 + 2*x^9 - 2*x^11 + 2*x^13 - 2*x^15 + ...
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MAPLE
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MATHEMATICA
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CoefficientList[Series[(1+x)^2/(1+x^2), {x, 0, 110}], x] (* or *) Join[ {1}, PadRight[{}, 120, {2, 0, -2, 0}]] (* Harvey P. Dale, Nov 23 2011 *)
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PROG
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(PARI) {a(n) = if( n<1, n==0, 2 * if( n%2, (-1)^(n\2)) )}; /* Michael Somos, Aug 04 2009 */
(Magma) [1] cat [Integers()!((1-(-1)^n)*(-1)^(n*(n-1)/2)): n in [1..100]]; // Wesley Ivan Hurt, Oct 27 2015
(PARI) a(n) = if(n==0, 1, I*((-I)^n-I^n)) \\ Colin Barker, Oct 27 2015
(PARI) Vec((1+x)^2/(1+x^2) + O(x^100)) \\ Colin Barker, Oct 27 2015
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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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