A154635 Ratio of the sum of the bends of the 5-dimensional spheres added in the n-th generation of Apollonian packing to the sum of the bends of the initial configuration of seven mutually tangent spheres.

1, 2, 15, 108, 774, 5544, 39708, 284400, 2036952, 14589216, 104492016, 748400832, 5360254560, 38391631488, 274971524544, 1969422407424, 14105550112128, 101027866452480, 723589630947072, 5182549848861696, 37118861005211136, 265855588948518912

Offset

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Colin Mallows, Growing Apollonian packings, J. Integer Sequences v.12, article 09.2.1 (2009).

Index entries for linear recurrences with constant coefficients, signature (8,-6).

Formula

G.f. (1-x)*(1-5*x) / (1-8*x+6*x^2).

From Colin Barker, Nov 16 2016: (Start)

a(n) = (((4-sqrt(10))^n*(-8+sqrt(10))+(4+sqrt(10))^n*(8+sqrt(10))))/(12*sqrt(10)) for n>0.

a(n) = 8*a(n-1) - 6*a(n-2) for n>2.

(End)

Example

Starting with seven 5-dimensional spheres with bends 0,0,1,1,1,1,1 summing to 5, the first derived generation has seven spheres, with bends 1,1,1,1,1,5/2,5/2 summing to 10. So a(1) = 10/5 = 2.

Mathematica

CoefficientList[Series[(1 - z) (1 - 5 z)/(1 - 8 z + 6 z^2), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)

Prog

(PARI) Vec((1-x)*(1-5*x)/(1-8*x+6*x^2) + O(x^30)) \\ Colin Barker, Nov 16 2016

Crossrefs

Cf. A135849 for dim=2. A137146 for the sum of squares of bends when dim=2. A154636 and A154637 for starting with three spheres in 2 dimensions. A154638-A154645 for results in the three-dimensional case.

Sequence in context: A279087 A037740 A037635 * A062808 A162773 A140637

Adjacent sequences: A154632 A154633 A154634 * A154636 A154637 A154638

Keyword

easy,nonn

Author

Colin Mallows, Jan 13 2009

Status

approved