%I #26 Sep 08 2022 08:44:59
%S 1,0,1,3,2,7,13,18,41,71,122,239,421,762,1417,2543,4642,8495,15389,
%T 28082,51177,93047,169610,308847,562197,1024170,1864841,3395711,
%U 6184498,11261551,20507789,37346914,68008809,123848199,225535258
%N Expansion of (1-x)/(1-x-x^2-2*x^3+2*x^4).
%H G. C. Greubel, <a href="/A052546/b052546.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=482">Encyclopedia of Combinatorial Structures 482</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,2,-2).
%F G.f.: (1-x)/(1-x-2*x^3+2*x^4-x^2).
%F a(n) = a(n-1) + a(n-2) + 2*a(n-3) - 2*a(n-4), with a(0)=1, a(1)=0, a(2)=1, a(3)=3.
%F a(n) = Sum_{alpha = RootOf(1-x-2*x^3+2*x^4-x^2)} (-1/353 * (-18-106*alpha+33*alpha^2+28*alpha^3) * alpha^(-1-n)).
%p spec := [S,{S=Sequence(Prod(Z,Z,Union(Z,Z,Sequence(Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..40);
%t CoefficientList[Series[(1-x)/(1-x-x^2-2x^3+2x^4),{x,0,40}],x] (* or *) LinearRecurrence[{1,1,2,-2},{1,0,1,3},40] (* _Harvey P. Dale_, Jul 02 2017 *)
%o (PARI) my(x='x+O('x^40)); Vec((1-x)/(1-x-2*x^3+2*x^4-x^2)) \\ _G. C. Greubel_, May 08 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)/(1-x-2*x^3+2*x^4-x^2) )); // _G. C. Greubel_, May 08 2019
%o (Sage) ((1-x)/(1-x-2*x^3+2*x^4-x^2)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, May 08 2019
%o (GAP) a:=[1,0,1,3];; for n in [5..40] do a[n]:=a[n-1]+a[n-2]+2*a[n-3] -2*a[n-4]; od; a; # _G. C. Greubel_, May 08 2019
%K easy,nonn
%O 0,4
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _James A. Sellers_, Jun 06 2000