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

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• sign

Comments

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

Links

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

Programs

Mathematica

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 *)

Formula

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