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%I #24 Sep 08 2022 08:44:59
%S 2,2,4,9,22,56,145,378,988,2585,6766,17712,46369,121394,317812,832041,
%T 2178310,5702888,14930353,39088170,102334156,267914297,701408734,
%U 1836311904,4807526977,12586269026,32951280100,86267571273
%N Expansion of (2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2)).
%H Vincenzo Librandi, <a href="/A052925/b052925.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=910">Encyclopedia of Combinatorial Structures 910</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).
%F G.f.: (2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2)).
%F a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), with a(0)=2, a(1)=2, a(2)=4, a(3)=9.
%F a(n) = 1 + Sum_{alpha=RootOf(1-3*z+z^2)} (1/5)*(2-3*alpha)*alpha^(-1-n).
%p spec:=[S,{S=Union(Sequence(Z),Sequence(Prod(Sequence(Z),Sequence(Z),Z) ))}, unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p seq(coeff(series((2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2)), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Oct 17 2019
%t CoefficientList[Series[(-2+6*x-4*x^2+x^3)/(-1+x)/(1-3*x+x^2),{x,0,40}],x] (* _Vincenzo Librandi_, Jun 22 2012 *)
%t LinearRecurrence[{4,-4,1}, {2,2,4,9}, 30] (* _G. C. Greubel_, Oct 17 2019 *)
%o (Magma) I:=[2,2,4,9]; [n le 4 select I[n] else 4*Self(n-1)-4*Self(n-2) +Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jun 22 2012
%o (PARI) my(x='x+O('x^30)); Vec((2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2))) \\ _G. C. Greubel_, Oct 17 2019
%o (Sage)
%o def A052925_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2))).list()
%o A052925_list(30) # _G. C. Greubel_, Oct 17 2019
%o (GAP) a:=[2,4,9];; for n in [4..30] do a[n]:=4*a[n-1]-4*a[n-2]+a[n-3]; od; Concatenation([2], a); # _G. C. Greubel_, Oct 17 2019
%Y Apart from first term, same as A055588.
%K easy,nonn
%O 0,1
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _James A. Sellers_, Jun 05 2000
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