A052925 Expansion of (2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2)).

2, 2, 4, 9, 22, 56, 145, 378, 988, 2585, 6766, 17712, 46369, 121394, 317812, 832041, 2178310, 5702888, 14930353, 39088170, 102334156, 267914297, 701408734, 1836311904, 4807526977, 12586269026, 32951280100, 86267571273

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 910
• Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Crossrefs

Apart from first term, same as A055588.

Programs

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

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

Maple

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

Mathematica

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 *)
LinearRecurrence[{4,-4,1}, {2,2,4,9}, 30] (* G. C. Greubel, Oct 17 2019 *)

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

Sage

def A052925_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2))).list()
A052925_list(30) # G. C. Greubel, Oct 17 2019

Formula

G.f.: (2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2)).

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.

a(n) = 1 + Sum_{alpha=RootOf(1-3*z+z^2)} (1/5)*(2-3*alpha)*alpha^(-1-n).

Extensions

More terms from James Sellers, Jun 05 2000