%I #21 Sep 08 2022 08:45:14

%S 1,1,1,5,13,25,57,141,325,737,1713,3989,9213,21289,49321,114205,

%T 264245,611569,1415713,3276837,7584237,17554489,40632089,94046637,

%U 217679141,503840001,1166187409,2699251381,6247675357,14460848969,33471028105

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

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

%H G. C. Greubel, <a href="/A097117/b097117.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,4).

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

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

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

%t 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 *)

%t 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 *)

%o (PARI) my(x='x+O('x^30)); Vec((1-x)/((1-x)^2-4*x^3)) \\ _G. C. Greubel_, Jun 06 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/((1-x)^2-4*x^3) )); // _G. C. Greubel_, Jun 06 2019

%o (Sage) ((1-x)/((1-x)^2-4*x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 06 2019

%o (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

%K easy,nonn

%O 0,4

%A _Paul Barry_, Jul 25 2004

%E Edited by _N. J. A. Sloane_, Aug 14 2008

%E Definition corrected by _Harvey P. Dale_, Jan 05 2019