A052927 Expansion of 1/(1-4*x-x^3).

1, 4, 16, 65, 264, 1072, 4353, 17676, 71776, 291457, 1183504, 4805792, 19514625, 79242004, 321773808, 1306609857, 5305681432, 21544499536, 87484608001, 355244113436, 1442520953280, 5857568421121, 23785517797920, 96584592144960, 392195937000961

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

A transform of A000302 under the mapping mapping g(x) -> (1/(1-x^3)) * g(x/(1-x^3)). - Paul Barry, Oct 20 2004

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 913
• 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 (4,0,1).

Crossrefs

Cf. A099503.

Programs

GAP

a:=[1,4,16];; for n in [4..30] do a[n]:=4*a[n-1]+a[n-3]; od; a; # G. C. Greubel, Oct 17 2019

Magma

I:=[1, 4, 16]; [n le 3 select I[n] else 4*Self(n-1)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012

Magma

R:=PowerSeriesRing(Integers(), 25); Coefficients(R!( 1/(1-4*x-x^3))); // Marius A. Burtea, Oct 18 2019

Maple

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

Mathematica

CoefficientList[Series[1/(1-4x-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{4,0,1},{1,4,16},40] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)

PARI

my(x='x+O('x^30)); Vec(1/(1-4*x-x^3)) \\ G. C. Greubel, Oct 17 2019

Sage

def A052927_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-4*x-x^3) ).list()
A052927_list(30) # G. C. Greubel, Oct 17 2019

Formula

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

a(n) = 4*a(n-1) + a(n-3), with a(0)=1, a(1)=4, a(2)=16.

a(n) = Sum_{r=RootOf(-1+4*z+z^3)} (1/283)*(64 + 9*r + 24*r^2)*r^(-1-n).

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k)*4^(n-3*k). - Paul Barry, Oct 20 2004