A052916 - OEIS

%I #27 Sep 08 2022 08:44:59

%S 1,0,0,2,1,1,5,5,6,15,20,27,51,76,110,185,286,430,690,1077,1651,2601,

%T 4065,6290,9841,15370,23885,37277,58176,90576,141245,220320,343296,

%U 535210,834605,1300877,2028001,3162001,4929150,7684275,11980276

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

%H G. C. Greubel, <a href="/A052916/b052916.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=899">Encyclopedia of Combinatorial Structures 899</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-1).

%F G.f.: (1-x)/(1 - x - 2*x^3 + x^4).

%F a(n) = a(n-1) + 2*a(n-3) - a(n-4), with a(0)=1, a(1)=0, a(2)=0, a(3)=2.

%F a(n) = Sum_{alpha=RootOf(1-z-2*z^3+z^4)} (1/643)*(-13 + 201*alpha - 38*alpha^2 - 18*alpha^3)*alpha^(-1-n).

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

%p seq(coeff(series((1-x)/(1-x-2*x^3+x^4), x, n+1), x, n), n = 0..50); # _G. C. Greubel_, Oct 16 2019

%t LinearRecurrence[{1,0,2,-1},{1,0,0,2},50]  (* _Harvey P. Dale_, Apr 21  2011 *)

%o (PARI) my(x='x+O('x^50)); Vec((1-x)/(1-x-2*x^3+x^4)) \\ _G. C. Greubel_, Oct 16 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x-2*x^3+x^4) )); // _G. C. Greubel_, Oct 16 2019

%o (Sage)

%o def A052916_list(prec):

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

%o     return P((1-x)/(1-x-2*x^3+x^4)).list()

%o A052916_list(50) # _G. C. Greubel_, Oct 16 2019

%o (GAP) a:=[1,0,0,2];; for n in [5..50] do a[n]:=a[n-1]+2*a[n-3]-a[n-4]; od; a; # _G. C. Greubel_, Oct 16 2019

%K easy,nonn

%O 0,4

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

%E More terms from _James A. Sellers_, Jun 05 2000