A052946 Expansion of (1-x)^2/(1-3*x+2*x^3-x^4).

1, 1, 4, 10, 29, 80, 224, 624, 1741, 4855, 13541, 37765, 105326, 293751, 819264, 2284905, 6372539, 17772840, 49567974, 138243749, 385558106, 1075311210, 2999014106, 8364169855, 23327445251, 65059618751, 181449530649

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1005
• Index entries for linear recurrences with constant coefficients, signature (3,0,-2,1).

Programs

GAP

a:=[1,1,4,10];; for n in [5..30] do a[n]:=3*a[n-1]-2*a[n-3]+a[n-4]; od; a; # G. C. Greubel, Oct 21 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)^2/(1-3*x+2*x^3-x^4) )); // G. C. Greubel, Oct 21 2019

Maple

spec := [S,{S=Sequence(Prod(Union(Prod(Sequence(Z),Sequence(Z)),Z),Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series((1-x)^2/(1-3*x+2*x^3-x^4), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 21 2019

Mathematica

LinearRecurrence[{3,0,-2,1}, {1,1,4,10}, 30] (* G. C. Greubel, Oct 21 2019 *)
CoefficientList[Series[(1-x)^2/(1-3x+2x^3-x^4),{x,0,30}],x] (* Harvey P. Dale, Aug 30 2020 *)

PARI

my(x='x+O('x^30)); Vec((1-x)^2/(1-3*x+2*x^3-x^4)) \\ G. C. Greubel, Oct 21 2019

Sage

def A052946_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)^2/(1-3*x+2*x^3-x^4)).list()
A052946_list(30) # G. C. Greubel, Oct 21 2019

Formula

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

a(n) = 3*a(n-1) - 2*a(n-3) + a(n-4).

a(n) = Sum_{alpha=RootOf(-1+3*z-2*z^3+z^4)} (1/643)*(79 + 128*alpha - 133*alpha^2 + 40*alpha^3)*alpha^(-1-n).

Extensions

More terms from James Sellers, Jun 06 2000