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A009116
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Expansion of e.g.f. cos(x) / exp(x).
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35
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1, -1, 0, 2, -4, 4, 0, -8, 16, -16, 0, 32, -64, 64, 0, -128, 256, -256, 0, 512, -1024, 1024, 0, -2048, 4096, -4096, 0, 8192, -16384, 16384, 0, -32768, 65536, -65536, 0, 131072, -262144, 262144, 0, -524288, 1048576, -1048576, 0, 2097152, -4194304
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OFFSET
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0,4
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COMMENTS
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Apart from signs, generated by 1,1 position of H_2^n = [1,1;-1,1]^n; and a(n) = 2^(n/2)*cos(Pi*n/2). - Paul Barry, Feb 18 2004
Equals binomial transform of "Period 4, repeat [1, 0, -1, 0]". - Gary W. Adamson, Mar 25 2009
Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012
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LINKS
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FORMULA
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a(n) = -2 * (a(n-1) + a(n-2)); a(0)=1, a(1)=-1. - Michael Somos, Nov 17 2002
G.f.: (1 + x) / (1 + 2*x + 2*x^2).
E.g.f.: cos(x) / exp(x).
a(n) = -4*a(n-4).
E.g.f.: cos(x)/exp(x) = 1 - x/(G(0)+1), where G(k) = 4k+1-x+(x^2)*(4k+1)/((2k+1)*(4k+3)-(x^2)+x*(2k+1)*(4k+3)/( 2k+2-x+x*(2k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
a(n) = (-1)^floor((n+1)/2)*2^(n-1)*H(n, n mod 2, 1/2) for n >= 3 where H(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], 2). - Peter Luschny, Sep 03 2019
a(n) = 2^(n/2)*ChebyshevT(n, -1/sqrt(2)). - G. C. Greubel, Apr 17 2023
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EXAMPLE
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G.f. = 1 - x + 2*x^3 - 4*x^4 + 4*x^5 - 8*x^7 + 16*x^8 - 16*x^9 + 32*x^11 - 64*x^12 + ...
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MAPLE
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A009116 := n->add((-1)^j*binomial(n, 2*j), j=0..floor(n/2));
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MATHEMATICA
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n = 50; (* n = 2 mod 4 *) (CoefficientList[ Series[ Cos[x]/Exp[x], {x, 0, n}], x]* Table[k!, {k, 0, n-1}] )[[1 ;; 45]] (* Jean-François Alcover, May 18 2011 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( (1 + x) / (1 + 2*x + 2*x^2) + x * O(x^n), n))} /* Michael Somos, Nov 17 2002 */
(Magma) R<x>:=PowerSeriesRing(Rationals(), 50); Coefficients(R!(Laplace( Exp(-x)*Cos(x) ))); // G. C. Greubel, Jul 22 2018; Apr 17 2023
(SageMath)
def A009116(n): return 2^(n/2)*chebyshev_T(n, -1/sqrt(2))
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CROSSREFS
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(With different signs) row sums of triangle A104597.
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KEYWORD
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sign,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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