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

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This sequence a(n) appears in the formula for the nonnegative powers of the algebraic number rho(9) := 2*cos(piPi/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)

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)

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<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,1).

Recurrence: a(1)=0, a(0n) =1, 3*a(n-2)=3, + a(n)+-3*), with a(n+0)=1)-, a(n+31)=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), _alpha=RootOf(-1+3*_Z^2+_Z^3)).

seq(coeff(series(1/(1-3*x^2-x^3), x, n+1), x, n), n = 0..40); # G. C. Greubel, Oct 17 2019

(PARI) x='x+O('x^9940); Vec(1/(1-3*x^2-x^3)) \\ Altug Alkan, Feb 20 2018

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

(Sage)

def A052931_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P(1/(1-3*x^2-x^3)).list()

A052931_list(40) # G. C. Greubel, Oct 17 2019

(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

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N. Gogin and A. Mylläri, <a href="http://math.unm.edu/~aca/ACA/2013/Nonstandard/Gogin.pdf">Padovan-like sequences and Bell polynomials</a>, Proceedings of Applications of Computer Algebra ACA, 2013.

<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,1)

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