This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A097117 #21 Sep 08 2022 08:45:14
%S A097117 1,1,1,5,13,25,57,141,325,737,1713,3989,9213,21289,49321,114205,
%T A097117 264245,611569,1415713,3276837,7584237,17554489,40632089,94046637,
%U A097117 217679141,503840001,1166187409,2699251381,6247675357,14460848969,33471028105
%N A097117 Expansion of (1-x)/((1-x)^2 - 4*x^3).
%C A097117 Related to the Lorenz-Poincaré geometry of the group PSL[2,C]. - _Roger L. Bagula_, Feb 17 2006
%H A097117 G. C. Greubel, <a href="/A097117/b097117.txt">Table of n, a(n) for n = 0..1000</a>
%H A097117 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,4).
%F A097117 G.f.: (1-x)/(1 - 2*x + x^2 - 4*x^3).
%F A097117 a(n) = 2*a(n-1) - a(n-2) + 4*a(n-3).
%F A097117 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, 2*k)*4^k.
%t A097117 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 A097117 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 A097117 (PARI) my(x='x+O('x^30)); Vec((1-x)/((1-x)^2-4*x^3)) \\ _G. C. Greubel_, Jun 06 2019
%o A097117 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/((1-x)^2-4*x^3) )); // _G. C. Greubel_, Jun 06 2019
%o A097117 (Sage) ((1-x)/((1-x)^2-4*x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 06 2019
%o A097117 (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 A097117 easy,nonn
%O A097117 0,4
%A A097117 _Paul Barry_, Jul 25 2004
%E A097117 Edited by _N. J. A. Sloane_, Aug 14 2008
%E A097117 Definition corrected by _Harvey P. Dale_, Jan 05 2019