A052931 Expansion of 1/(1 - 3*x^2 - x^3).

1, 0, 3, 1, 9, 6, 28, 27, 90, 109, 297, 417, 1000, 1548, 3417, 5644, 11799, 20349, 41041, 72846, 143472, 259579, 503262, 922209, 1769365, 3269889, 6230304, 11579032, 21960801, 40967400, 77461435, 144863001, 273351705, 512050438, 964918116, 1809503019

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Let A be the tridiagonal unit-primitive matrix (see [Jeffery]) A = A_{9,1} = [0,1,0,0; 1,0,1,0; 0,1,0,1; 0,0,1,1]. Then a(n)=[A^n](2,3). - _L. Edson Jeffery, Mar 19 2011

From Wolfdieter Lang, Oct 02 2013: (Start)

This sequence a(n) appears in the formula for the nonnegative powers of the algebraic number rho(9) := 2*cos(Pi/9) of degree 3, the ratio of the smallest diagonal/side in the regular 9-gon, in terms of the power basis of the algebraic number field Q(rho(9)) (see A187360, n=9).

rho(9)^n = A(n)*1 + B(n)*rho(9) + C(n)*rho(9)^2, with A(0) = 1, A(1) = 0, A(n) = B(n-2), n >= 2, B(0) = 0, B(n) = a(n-1), n >= 1, C(0) = 0, C(n) = B(n-1), n >= 1. (End)

Examples

			From _Wolfdieter Lang_, Oct 02 2013: (Start)
In the 9-gon (enneagon), powers of rho(9) = 2*cos(pi/9):
rho(9)^5 = A(5)*1 + B(5)*rho(9) + C(5)*rho(9)^2, with A(5) = B(3) = a(2) = 3, B(5) = a(4) = 9 and C(5) = B(4) = a(3) = 1:
  rho(9)^5 = 3 + 9*rho(9) + rho(9)^2. (End)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• N. Gogin and A. Mylläri, Padovan-like sequences and Bell polynomials, Proceedings of Applications of Computer Algebra ACA, 2013.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 917
• L. E. Jeffery, Unit-primitive matrices
• Index entries for linear recurrences with constant coefficients, signature (0,3,1).

Crossrefs

Cf. A214699.

Programs

GAP

a:=[1,0,3];; for n in [4..40] do a[n]:=3*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Oct 17 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-3*x^2-x^3) )); // G. C. Greubel, Oct 17 2019

Maple

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

Mathematica

CoefficientList[Series[1/(1-3x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{0,3,1},{1,0,3},40] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)

PARI

x='x+O('x^40); Vec(1/(1-3*x^2-x^3)) \\ Altug Alkan, Feb 20 2018

Sage

def A052931_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/(1-3*x^2-x^3)).list()
A052931_list(40) # G. C. Greubel, Oct 17 2019

Formula

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

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

a(n) = Sum_{alpha=RootOf(-1+3*z^2+z^3)} (1/9)*(-1 +5*alpha +2*alpha^2) * alpha^(-1-n).

a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)3^(3k-n). - Paul Barry, Oct 04 2004

a(n) = A187497(3*(n+1)). - L. Edson Jeffery, Mar 19 2011.

3*a(n) = abs(A214699(n+1)). - Roman Witula, Oct 06 2012

Extensions

More terms from James Sellers, Jun 06 2000