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A052931 Expansion of 1/(1 - 3*x^2 - x^3). 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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)

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)

FORMULA

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

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

Sum(1/9*(-1+5*_alpha+2*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(-1+3*_Z^2+_Z^3))

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

EXAMPLE

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)

MAPLE

spec := [S, {S=Sequence(Prod(Z, Union(Z, Z, Z, Prod(Z, Z))))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

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 *)

PROG

(PARI) x='x+O('x^99); Vec(1/(1-3*x^2-x^3)) \\ Altug Alkan, Feb 20 2018

CROSSREFS

Cf. A214699.

Sequence in context: A127552 A229759 A185580 * A006803 A197730 A231902

Adjacent sequences:  A052928 A052929 A052930 * A052932 A052933 A052934

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

More terms from James A. Sellers, Jun 06 2000

STATUS

approved

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Last modified March 25 12:11 EDT 2019. Contains 321470 sequences. (Running on oeis4.)