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A108520 Expansion of 1/(1+2*x+2*x^2). 17
1, -2, 2, 0, -4, 8, -8, 0, 16, -32, 32, 0, -64, 128, -128, 0, 256, -512, 512, 0, -1024, 2048, -2048, 0, 4096, -8192, 8192, 0, -16384, 32768, -32768, 0, 65536, -131072, 131072, 0, -262144, 524288, -524288, 0, 1048576, -2097152, 2097152, 0, -4194304, 8388608, -8388608 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Yet another variation on A009545.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Maran van Heesch, The multiplicative complexity of symmetric functions over a field with characteristic p, Thesis, 2014.

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

FORMULA

G.f.: 1/(1+2*x+2*x^2).

E.g.f.: exp(-x)*(cos(x)-sin(x)).

a(n) = -2*(a(n-1)+a(n-2)).

a(n) = sum{k=0..n, sum{j=0..n-k, C(k,j)C(k,n-j)(-2)^(n-j)}} - Paul Barry, Mar 09 2006

a(n) = -4 * a(n-4). - Paul Curtz, Apr 24 2011

a(n) = A016116(n+1) * A075553(n+1). - Paul Curtz, Apr 25 2011

a(n) = -(-1-i)^(n-1)-(-1+i)^(n-1), where i=sqrt(-1). - Bruno Berselli, Apr 26 2011

a(n) = -2*A009116(n-1) for n>0. - Bruno Berselli, Apr 26 2011

Imaginary part of (-1+i)^n, negated real part is A090132. [Joerg Arndt, May 13 2011]

E.g.f.: (cos(x)-sin(x))*exp(-x)=G(0); G(k)=1-2*x/(4*k+1+x*(4*k+1)/(2*(2*k+1)-x-2*(x^2)*(2*k+1)/((x^2)-(2*k+2)*(4*k+3)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011

G.f.: G(0)/(2*(1 + x)), 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) = (-2)^n*hypergeom([1/2-n/2, -n/2], [-n], 2)) for n>=1. - Peter Luschny, Dec 17 2015

MAPLE

A108520 := n -> `if`(n=0, 1, (-2)^n*hypergeom([1/2-n/2, -n/2], [-n], 2)):

seq(simplify(A108520(n)), n=0..46); # Peter Luschny, Dec 17 2015

MATHEMATICA

CoefficientList[Series[1/(1 + 2 x + 2 x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{-2, -2}, {1, -2}, 50] (* Harvey P. Dale, Sep 30 2012 *)

Table[-(-1 - I)^(n - 1) - (-1 + I)^(n - 1), {n, 0, 50}] (* Bruno Berselli, Nov 08 2015 *)

PROG

(PARI) a(n)=if(n<0, 0, polcoeff(1/(1+2*x+2*x^2)+x*O(x^n), n))

(PARI) a(n)=if(n<1, n==0, -polsym(2+2*x+x^2, n-1)[n])

(MAGMA) [n le 2 select n*(-1)^(n-1) else -2*(Self(n-1)+Self(n-2)): n in [1..47]];  // Bruno Berselli, Apr 26 2011

(PARI) vector(66, n, imag((-1+I)^n)) /* Joerg Arndt, May 13 2011 */

CROSSREFS

Cf. a(n)=(-1)^n A099087(n). a(n)=-A084102(n) if n>0.

Cf. A009116.

Sequence in context: A180813 A194656 A283240 * A099087 A009545 A084102

Adjacent sequences:  A108517 A108518 A108519 * A108521 A108522 A108523

KEYWORD

sign,easy

AUTHOR

Michael Somos, Jun 07 2005

STATUS

approved

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Last modified March 24 22:17 EDT 2017. Contains 284035 sequences.