A004291 Expansion of (1 + 2*x + x^2)/(1 - 10*x + x^2).

1, 12, 120, 1188, 11760, 116412, 1152360, 11407188, 112919520, 1117788012, 11064960600, 109531817988, 1084253219280, 10733000374812, 106245750528840, 1051724504913588, 10410999298607040, 103058268481156812, 1020171685512961080, 10098658586648453988

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• J. M. Alonso, Growth functions of amalgams, in Alperin, ed., Arboreal Group Theory, Springer, pp. 1-34, esp. p. 32.
• P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 160, middle display.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Hacène Belbachir, Soumeya Merwa Tebtoub, László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
• Index entries for linear recurrences with constant coefficients, signature (10,-1).

Crossrefs

Pairwise sums of A054320.

Programs

Mathematica

CoefficientList[Series[(1+2*x+x^2)/(1-10*x+x^2),{x,0,30}],x] (* Vincenzo Librandi, Jun 13 2012 *)

PARI

Vec((1+2*x+x^2)/(1-10*x+x^2) + O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = 12*A004189(n), n> 0. - R. J. Mathar, Oct 29 2012

a(n) = sqrt(3/2)*(-(5 - 2*sqrt(6))^n + (5 + 2*sqrt(6))^n) for n > 0. - Colin Barker, Jan 25 2016

For n > 0: a(n) = 10*a(n-1) - a(n-2) with a(0) = 0, a(1) = 12. - A.H.M. Smeets, Jul 25 2017