A052949 Expansion of (2-4*x+x^3)/((1-x)*(1-2*x-x^2+x^3)).

2, 2, 4, 7, 15, 32, 71, 158, 354, 794, 1783, 4005, 8998, 20217, 45426, 102070, 229348, 515339, 1157955, 2601900, 5846415, 13136774, 29518062, 66326482, 149034251, 334876921, 752461610, 1690765889, 3799116466, 8536537210, 19181424996

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 1008
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-2,1).

Crossrefs

Cf. A006356.

Programs

GAP

a:=[2,2,4,7];; for n in [5..40] do a[n]:=3*a[n-1]-a[n-2]-2*a[n-3] +a[n-4]; od; a; # G. C. Greubel, Oct 21 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-4*x+x^3)/((1-x)*(1-2*x-x^2+x^3)) )); // G. C. Greubel, Oct 21 2019

Maple

spec:= [S,{S=Union(Sequence(Prod(Union(Sequence(Z),Z),Z)),Sequence(Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series((2-4*x+x^3)/((1-x)*(1-2*x-x^2+x^3)), x, n+1), x, n), n = 0..40); # G. C. Greubel, Oct 21 2019

Mathematica

LinearRecurrence[{3,-1,-2,1}, {2,2,4,7}, 40] (* G. C. Greubel, Oct 21 2019 *)
CoefficientList[Series[(2-4x+x^3)/((1-x)(1-2x-x^2+x^3)),{x,0,50}],x] (* Harvey P. Dale, Jul 30 2024 *)

PARI

my(x='x+O('x^40)); Vec((2-4*x+x^3)/((1-x)*(1-2*x-x^2+x^3))) \\ G. C. Greubel, Oct 21 2019

Sage

def A052949_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((2-4*x+x^3)/((1-x)*(1-2*x-x^2+x^3))).list()
A052949_list(40) # G. C. Greubel, Oct 21 2019

Formula

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

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

a(n) = A006356(n-1) + 1, n>0.

a(n) = 1 + Sum_{alpha=RootOf(1-2*z-z^2+z^3)} (1/7)*(1 + 2*alpha - alpha^2)*alpha^(-1-n).

Extensions

More terms from James Sellers, Jun 05 2000