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A097073 Expansion of (1-x+2*x^2)/((1+x)*(1-2*x)). 23
1, 0, 4, 4, 12, 20, 44, 84, 172, 340, 684, 1364, 2732, 5460, 10924, 21844, 43692, 87380, 174764, 349524, 699052, 1398100, 2796204, 5592404, 11184812, 22369620, 44739244, 89478484, 178956972, 357913940, 715827884, 1431655764, 2863311532 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Partial sums are A097074. Pairwise sums are {1,1,4,16,32,...} or 2^n-sum(k=0..n, binomial(n,k)*(-1)^(n+k)*k).

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (1,2).

FORMULA

a(n) = 2*2^n/3+4(-1)^n/3-0^n.

a(n) = A001045(n+1)+(-1)^n-0^n.

a(n) = 2*A078008(n)-0^n.

a(2*n+1)+a(2*n+2) = A000302(n+1). - Paul Curtz, Jun 30 2008

G.f.: 1 - x + x*Q(0), where Q(k) = 1 + 2*x^2 + (4*k+5)*x - x*(4*k+1 + 2*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013

MATHEMATICA

k=0; lst={1, k}; Do[k=2^n-k; AppendTo[lst, k], {n, 2, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)

CoefficientList[Series[(1-x+2x^2)/((1+x)(1-2x)), {x, 0, 40}], x] (* Harvey P. Dale, Dec 10 2012 *)

PROG

(MAGMA) [2*2^n/3+4*(-1)^n/3-0^n: n in [0..35]]; // Vincenzo Librandi, Aug 12 2011

(PARI) a(n)=([0, 1; 2, 1]^n*[1; 0])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016

CROSSREFS

Cf. A046055.

Cf. A001045, A078008 (form a(n)=2^n-a(n-1)).

Sequence in context: A053415 A079902 A120033 * A019085 A106232 A228612

Adjacent sequences:  A097070 A097071 A097072 * A097074 A097075 A097076

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jul 22 2004

EXTENSIONS

Obscure variable k in Orlovsky comment replaced with a(n) by R. J. Mathar, Apr 23 2009

STATUS

approved

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Last modified November 21 13:53 EST 2017. Contains 295001 sequences.