A078026 Expansion of (1-x)/(1-x^2-2*x^3).

1, -1, 1, 1, -1, 3, 1, 1, 7, 3, 9, 17, 15, 35, 49, 65, 119, 163, 249, 401, 575, 899, 1377, 2049, 3175, 4803, 7273, 11153, 16879, 25699, 39185, 59457, 90583, 137827, 209497, 318993, 485151, 737987, 1123137, 1708289, 2599111, 3954563, 6015689, 9152785, 13924815

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• sign
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Programs

GAP

a:=[1,-1,1];; for n in [4..50] do a[n]:=a[n-2]+2*a[n-3]; od; a; # G. C. Greubel, Aug 04 2019

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x^2-2*x^3) )); // G. C. Greubel, Aug 04 2019

Maple

seq(coeff(series((1-x)/(-2*x^3-x^2+1), x, n+1), x, n), n = 0..50); # G. C. Greubel, Aug 04 2019

Mathematica

CoefficientList[Series[(1-x)/(1-x^2-2*x^3), {x,0,50}], x] (* Wesley Ivan Hurt, Jan 24 2017 *)

Maxima

a(n):=sum(sum(binomial(m+i-1,m-1)*sum(binomial(j,n-3*m+2*j-i)*2^(m-j)*binomial(m,j)*(-1)^(-n+3*m-j+i),j,0,m),i,0,n-m),m,1,n); /* Vladimir Kruchinin, May 12 2011 */

PARI

Vec((1-x)/(1-x^2-2*x^3) + O(x^50)) \\ Felix Fröhlich, Jan 24 2017

Sage

((1-x)/(1-x^2-2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Aug 04 2019

Formula

a(n) = Sum_{m=1..n} Sum_{i=0..n-m} binomial(m+i-1,m-1)*Sum_{j=0..m} binomial(j,n-3*m+2*j-i)*2^(m-j)*binomial(m,j)*(-1)^(-n+3*m-j+i), n > 0. - Vladimir Kruchinin, May 12 2011