A009116 Expansion of e.g.f. cos(x) / exp(x).

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

Offset

0,4

Comments

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

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..2000

Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.

N. J. A. Sloane, Table of n, (I-1)^n for n=0..100

Index entries for linear recurrences with constant coefficients, signature (-2,-2).

Formula

Real part of (-1-i)^n. See A009545 for imaginary part. - Marc LeBrun

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) = Sum_{k=0..n} (-1)^k*A098158(n,k). - Philippe Deléham, Dec 04 2006

a(n)*(-1)^n = A099087(n) - A099087(n-1). - R. J. Mathar, Nov 18 2007

a(n) = (-1)^n*A146559(n). - Philippe Deléham, Dec 01 2008

From Paul Curtz, Jul 22 2011: (Start)

a(n) = -4*a(n-4).

a(n) = A016116(n) * A075553(n+6). (End)

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)^n*2^(n/2)*cos(n*Pi/4). - Nordine Fahssi, Dec 18 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

a(n) = A108520(n-1)+A108520(n). - R. J. Mathar, May 09 2023

Example

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 + ...

Maple

A009116 := n->add((-1)^j*binomial(n, 2*j), j=0..floor(n/2));

Mathematica

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 *)
Table[(1/2)*((-1-I)^n + (-1+I)^n), {n, 0, 50}] (* Jean-François Alcover, Jan 31 2018 *)

Prog

(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))
[A009116(n) for n in range(41)] # G. C. Greubel, Apr 17 2023

Crossrefs

Cf. A009545, A016116, A075553, A090132, A098158, A099087, A146559.

(With different signs) row sums of triangle A104597.

Also related to A066321 and A271472.

Sequence in context: A195479 A112793 A372037 * A146559 A118434 A090132

Adjacent sequences: A009113 A009114 A009115 * A009117 A009118 A009119

Keyword

sign,easy,nice

Author

R. H. Hardin

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Definition corrected by Joerg Arndt, Apr 29 2011

Status

approved