A097117 Expansion of (1-x)/((1-x)^2 - 4*x^3).

1, 1, 1, 5, 13, 25, 57, 141, 325, 737, 1713, 3989, 9213, 21289, 49321, 114205, 264245, 611569, 1415713, 3276837, 7584237, 17554489, 40632089, 94046637, 217679141, 503840001, 1166187409, 2699251381, 6247675357, 14460848969, 33471028105

Offset: 0

Author

Paul Barry, Jul 25 2004

Keywords

• easy
• nonn

Comments

Related to the Lorenz-Poincaré geometry of the group PSL[2,C]. - Roger L. Bagula, Feb 17 2006

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,4).

Programs

GAP

a:=[1,1,1];; for n in [4..30] do a[n]:=2*a[n-1]-a[n-2]+4*a[n-3]; od; a; # G. C. Greubel, Jun 06 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/((1-x)^2-4*x^3) )); // G. C. Greubel, Jun 06 2019

Mathematica

M = {{0, 1, 0}, {0, 0, 1}, {4, -1, 2}}; w[0] = {0, 1, 1}; w[n_] := w[n] = M.w[n - 1] a = Flatten[Table[w[n][[1]], {n, 0, 25}]] (* Roger L. Bagula, Feb 17 2006 *)
CoefficientList[Series[(1-x)/((1-x)^2-4x^3),{x,0,30}],x] (* or *) LinearRecurrence[{2,-1,4},{1,1,1},40] (* Harvey P. Dale, Jan 05 2019 *)

PARI

my(x='x+O('x^30)); Vec((1-x)/((1-x)^2-4*x^3)) \\ G. C. Greubel, Jun 06 2019

Sage

((1-x)/((1-x)^2-4*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 06 2019

Formula

G.f.: (1-x)/(1 - 2*x + x^2 - 4*x^3).

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

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, 2*k)*4^k.

Extensions

Edited by N. J. A. Sloane, Aug 14 2008

Definition corrected by Harvey P. Dale, Jan 05 2019