A052917 Expansion of 1/(1-3*x-x^4).

1, 3, 9, 27, 82, 249, 756, 2295, 6967, 21150, 64206, 194913, 591706, 1796268, 5453010, 16553943, 50253535, 152556873, 463123629, 1405924830, 4268028025, 12956640948, 39333046473, 119405064249, 362483220772, 1100406303264

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

a(n) equals the number of n-length words on {0,1,2,3} such that 0 appears only in a run whose length is a multiple of 4. - Milan Janjic, Feb 17 2015

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 900
• Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
• Index entries for linear recurrences with constant coefficients, signature (3,0,0,1).

Programs

GAP

a:=[1,3,9,27];; for n in [5..30] do a[n]:=3*a[n-1]+a[n-4]; od; a; # G. C. Greubel, Oct 16 2019

Magma

[n le 4 select 3^(n-1) else 3*Self(n-1)+Self(n-4): n in [1..30]]; // Vincenzo Librandi, Feb 20 2015

Magma

R:=PowerSeriesRing(Integers(), 27); Coefficients(R!( 1/(1-3*x-x^4) )); // Marius A. Burtea, Oct 16 2019

Maple

spec := [S,{S=Sequence(Union(Z,Z,Z,Prod(Z,Z,Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(coeff(series(x^4/((1+2*x)*(2*x^3+x^2-2*x+1)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 16 2019

Mathematica

CoefficientList[Series[1/(1-3x-x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 20 2015 *)
RecurrenceTable[{a[0]==1, a[1]==3, a[2]==9, a[3]==27, a[n]==3a[n-1] +a[n -4]}, a[n], {n, 0, 30}] (* Bruno Berselli, Feb 20 2015 *)

PARI

Vec(1/(1-3*x-x^4) + O(x^30)) \\ Michel Marcus, Feb 17 2015

Sage

def A052917_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/(1-3*x-x^4)).list()
A052917_list(30) # G. C. Greubel, Oct 16 2019

Formula

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

a(n) = 3*a(n-1) + a(n-4), with a(0)=1, a(1)=3, a(2)=9, a(3)=27.

a(n) = Sum_{alpha=RootOf(-1 + 3*z + z^4)} (1/2443)*(729 + 64*alpha + 144*alpha^2 + 324*alpha^3)*alpha^(-1-n).

Extensions

More terms from James Sellers, Jun 06 2000