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%I #21 Sep 08 2022 08:45:00
%S 1,3,13,53,219,903,3725,15365,63379,261431,1078373,4448165,18348171,
%T 75684103,312188253,1287740773,5311783139,21910496823,90378288885,
%U 372800086085,1537757639579,6343074066631,26164453733933,107925373723205,445179800493491
%N Expansion of ( 1-x ) / ( 1-4*x-x^2+2*x^3 ).
%H Vincenzo Librandi, <a href="/A052990/b052990.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1064">Encyclopedia of Combinatorial Structures 1064</a>
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (4,1,-2).
%F G.f.: (1-x)/(1-4*x-x^2+2*x^3)
%F Recurrence: {a(0)=1, a(1)=3, a(2)=13, 2*a(n)-a(n+1)-4*a(n+2)+a(n+3)=0}
%F Sum(-1/142*(-22+18*r^2-21*r)*r^(-1-n) where r=RootOf(1-4*_Z-_Z^2+2*_Z^3))
%p spec := [S,{S=Sequence(Union(Prod(Union(Sequence(Z),Z),Union(Z,Z)),Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%t LinearRecurrence[{4,1,-2},{1,3,13},40] (* _Vincenzo Librandi_, Jun 23 2012 *)
%t CoefficientList[Series[(1-x)/(1-4x-x^2+2x^3),{x,0,40}],x] (* _Harvey P. Dale_, Jan 15 2022 *)
%o (Magma) I:=[1, 3, 13]; [n le 3 select I[n] else 4*Self(n-1)+Self(n-2)-2*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jun 23 2012
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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