A063809 Spherical growth series for pair of groups, one Gromov hyperbolic, the other not.

1, 108, 2160, 40176, 743040, 13735872, 253912320, 4693641984, 86763294720, 1603843918848, 29647506124800, 548042492768256, 10130719683870720, 187269203879903232, 3461723926449684480, 63990940820356399104, 1182890546466536816640

Offset

0,2

Comments

Limit n -> infinity, a(n+1)/a(n) = 10+6*sqrt(2). - Avi Friedlich, Jun 02 2015

References

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 154.

Links

Table of n, a(n) for n=0..16.

W. Floyd and W. Parry, The growth of nonpositively curved triangles of groups, Invent. math., 129 (1997), 289-359.

Index entries for linear recurrences with constant coefficients, signature (20, -28).

Formula

G.f.: (1+88*x+28*x^2)/(1-20*x+28*x^2).

a(n) = 9*((10+6*sqrt(2))^n-(10-6*sqrt(2))^n)/sqrt(2), n>0. - Benedict W. J. Irwin, Jul 14 2016

E.g.f.: 1 + 9*sqrt(2)*sinh(6*sqrt(2)*x)*exp(10*x). - Ilya Gutkovskiy, Jul 14 2016

Mathematica

CoefficientList[Series[(1 + 88 x + 28 x^2)/(1 - 20 x + 28 x^2), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 02 2015 *)

Prog

(PARI) Vec((1+88*x+28*x^2)/(1-20*x+28*x^2) + O(x^30)) \\ Michel Marcus, Jun 02 2015

Crossrefs

Sequence in context: A250339 A263961 A202427 * A183357 A264681 A221018

Adjacent sequences: A063806 A063807 A063808 * A063810 A063811 A063812

Keyword

nonn

Author

N. J. A. Sloane, Aug 20 2001

Status

approved