A117373 -id:A117373 - cp's OEIS frontend

A077903 Expansion of (1-x)^(-1)/(1 + x - x^2 + 2*x^3).

1, 0, 2, -3, 6, -12, 25, -48, 98, -195, 390, -780, 1561, -3120, 6242, -12483, 24966, -49932, 99865, -199728, 399458, -798915, 1597830, -3195660, 6391321, -12782640, 25565282, -51130563, 102261126, -204522252, 409044505, -818089008, 1636178018, -3272356035, 6544712070

Offset: 0

N. J. A. Sloane, Nov 17 2002

sign

Convolution of A010892(n) and (-1)^n*A001045(n+1). The positive sequence has g.f. 1/((1-x-2x^2)*(1+x+x^2)). This is the convolution of A001045(n+1) and A049347(n). - Paul Barry, May 19 2004

Index entries for linear recurrences with constant coefficients, signature (0,2,-3,2)

CoefficientList[Series[(1-x)^(-1)/(1+x-x^2+2x^3),{x,0,40}],x] (* or *) LinearRecurrence[{0,2,-3,2},{1,0,2,-3},40] (* Harvey P. Dale, Apr 25 2016 *)

G.f.: 1/((1+x-2x^2)*(1-x+x^2));
a(n) = Sum_{k=0..n} (2*(-2)^k/3 + 1/3)*2*sin(Pi*(n-k)/3 + Pi/3)/sqrt(3);
a(n) = 2^(n+3)*cos(Pi*n)/21 + 8*sqrt(3)*cos(Pi*n/3 + Pi/6)/63 + 4*sqrt(3)*sin(Pi*n/3 + Pi/3)/63 + 2*sqrt(3)*sin(Pi*n/3)/9 + 1/3. - Paul Barry, May 19 2004
a(n) = 1/3 + (-1)^n*2^(n+3)/21 - A117373(n+1)/7. - R. J. Mathar, Sep 27 2012

A133511 a(n) = 3 A113405(n)- A113405(n+1).

Original entry on oeis.org

0, 0, -1, 1, 2, 5, 7, 14, 27, 57, 114, 229, 455, 910, 1819, 3641, 7282, 14565, 29127, 58254, 116507, 233017, 466034, 932069, 1864135, 3728270, 7456539, 14913081, 29826162, 59652325, 119304647, 238609294, 477218587, 954437177, 1908874354, 3817748709, 7635497415

Offset: 0

Paul Curtz, Nov 30 2007

sign

2a(n)-a(n+1)=A133513(n).
A113405(n)-a(n)=A131531(n).
O.g.f.: x^2(3x-1)/((1-2x)(1+x)(1-x+x^2)). - R. J. Mathar, Jul 22 2008
a(n+2)=4*(2^n-(-1)^n)/9+A117373(n+2)/3. [From R. J. Mathar, Jul 20 2009]

Edited and extended by R. J. Mathar, Jul 22 2008

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A077903 Expansion of (1-x)^(-1)/(1 + x - x^2 + 2*x^3).

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A133511 a(n) = 3 A113405(n)- A113405(n+1).

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