A052926 Expansion of (1-3*x)/(1 - 4*x - x^2 + 3*x^3).

1, 1, 5, 18, 74, 299, 1216, 4941, 20083, 81625, 331760, 1348416, 5480549, 22275332, 90536629, 367980201, 1495631437, 6078896062, 24707275082, 100421102079, 408154995212, 1658919257681, 6742568719699, 27404729150841

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 912

Index entries for linear recurrences with constant coefficients, signature (4,1,-3).

Formula

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

a(n) = 4*a(n-1) + a(n-2) - 3*a(n-3), with a(0)=1, a(1)=1, a(2)=5.

a(n) = Sum_{r=RootOf(1-4*z-z^2+3*z^3)} (-1/761)*(17 -278*r +15*r^2)*r^(-1-n).

Maple

spec:= [S, {S=Sequence(Prod(Z, Union(Z, Sequence(Union(Z, Z, Z))) ))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
seq(coeff(series((1-3*x)/(1-4*x-x^2+3*x^3), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 17 2019

Mathematica

LinearRecurrence[{4, 1, -3}, {1, 1, 5}, 40] (* Vincenzo Librandi, Jun 22 2012 *)

Prog

(Magma) I:=[1, 1, 5]; [n le 3 select I[n] else 4*Self(n-1)+Self(n-2) -3*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012
(PARI) my(x='x+O('x^30)); Vec((1-3*x)/(1-4*x-x^2+3*x^3)) \\ G. C. Greubel, Oct 17 2019
(Sage)
def A052926_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)/(1-4*x-x^2+3*x^3)).list()
A052926_list(30) # G. C. Greubel, Oct 17 2019
(GAP) a:=[1, 1, 5];; for n in [4..30] do a[n]:=4*a[n-1]+a[n-2]-3*a[n-3]; od; a; # G. C. Greubel, Oct 17 2019

Crossrefs

Sequence in context: A145780 A183443 A034551 * A296123 A242054 A027134

Adjacent sequences: A052923 A052924 A052925 * A052927 A052928 A052929

Keyword

easy,nonn

Author

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

Extensions

More terms from James A. Sellers, Jun 08 2000

Status

approved