A052541 Expansion of 1/(1-3*x-x^3).

1, 3, 9, 28, 87, 270, 838, 2601, 8073, 25057, 77772, 241389, 749224, 2325444, 7217721, 22402387, 69532605, 215815536, 669848995, 2079079590, 6453054306, 20029011913, 62166115329, 192951400293, 598883212792, 1858815753705, 5769398661408, 17907079197016

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

A transform of A000244 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} such that 0 appears only in a run which length is a multiple of 3. - 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 475
• Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
• José L. Ramírez, Víctor F. Sirvent, A note on the k-Narayana sequence, Annales Mathematicae et Informaticae, 45 (2015) pp. 91-105.
• Index entries for linear recurrences with constant coefficients, signature (3,0,1).

Crossrefs

Cf. A076264.

Programs

GAP

a:=[1,3,9];; for n in [4..30] do a[n]:=3*a[n-1]+a[n-3]; od; a; # G. C. Greubel, May 09 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x/(1-3*x-x^3) )); // G. C. Greubel, May 09 2019

Maple

spec := [S,{S=Sequence(Union(Z,Z,Z,Prod(Z,Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..30);

Mathematica

CoefficientList[Series[x/(1-3*x-x^3), {x, 0, 30}], x] (* Zerinvary Lajos, Mar 29 2007 *)
LinearRecurrence[{3,0,1},{1,3,9},30] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)

PARI

my(x='x+O('x^30)); Vec(x/(1-3*x-x^3)) \\ G. C. Greubel, May 09 2019

Sage

(x/(1-3*x-x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

Formula

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

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

a(n) = Sum_{alpha = RootOf(-1+3*x+x^3)} (1/15)*(4 + alpha + 2*alpha^2) * alpha^(-n-1).

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

Extensions

More terms from James Sellers, Jun 06 2000