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A052911 Expansion of (1-x)/(1 - 3*x - x^2 + 2*x^3). 4

%I

%S 1,2,7,21,66,205,639,1990,6199,19309,60146,187349,583575,1817782,

%T 5662223,17637301,54938562,171128541,533049583,1660400166,5171992999,

%U 16110279997,50182032658,156312391973,486898648583,1516644272406

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

%H G. C. Greubel, <a href="/A052911/b052911.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=891">Encyclopedia of Combinatorial Structures 891</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-2).

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

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

%F a(n) = Sum_{alpha=RootOf(1 - 3*z - z^2 + 2*z^3)} (1/229)*(43 + 41*alpha - 46*alpha^2)*alpha^(-1-n).

%F a(n) = center term in M^n * [1 1 1] where M = Hosoya's triangle considered as an upper triangular 3 X 3 matrix: [2 1 2 / 1 1 0 / 1 0 0]. E.g., a(4) = 66 since M^4 * [1 1 1] = [139 66 45]. The analogous procedure using M^n * [1 0 0] generates A100058. - _Gary W. Adamson_, Oct 31 2004

%F a(n) = A100058(n) - A100058(n-1). - _R. J. Mathar_, May 04 2018

%p spec := [S,{S=Sequence(Union(Z,Prod(Union(Sequence(Z),Z,Z),Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t LinearRecurrence[{3,1,-2}, {1,2,7}, 30] (* _G. C. Greubel_, Oct 15 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec((1-x)/(1-3*x-x^2+2*x^3)) \\ _G. C. Greubel_, Oct 15 2019

%o (MAGMA) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-3*x-x^2+2*x^3) )); // _G. C. Greubel_, Oct 15 2019

%o (Sage)

%o def A052911_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P((1-x)/(1-3*x-x^2+2*x^3)).list()

%o A052911_list(30) # _G. C. Greubel_, Oct 15 2019

%o (GAP) a:=[1,2,7];; for n in [4..30] do a[n]:=3*a[n-1]+a[n-2]-2*a[n-3]; od; a; # _G. C. Greubel_, Oct 15 2019

%Y Cf. A100058, A058071, A100059.

%K easy,nonn

%O 0,2

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E More terms from _James A. Sellers_, Jun 06 2000

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Last modified February 27 15:59 EST 2020. Contains 332307 sequences. (Running on oeis4.)