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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.

%I A154635 #13 Nov 17 2016 23:37:56
%S A154635 1,2,15,108,774,5544,39708,284400,2036952,14589216,104492016,
%T A154635 748400832,5360254560,38391631488,274971524544,1969422407424,
%U A154635 14105550112128,101027866452480,723589630947072,5182549848861696,37118861005211136,265855588948518912
%N 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.
%H A154635 Colin Barker, <a href="/A154635/b154635.txt">Table of n, a(n) for n = 0..1000</a>
%H A154635 Colin Mallows, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Mallows/mallows8.html">Growing Apollonian packings</a>, J. Integer Sequences v.12, article 09.2.1 (2009).
%H A154635 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-6).
%F A154635 G.f. (1-x)*(1-5*x) / (1-8*x+6*x^2).
%F A154635 From _Colin Barker_, Nov 16 2016: (Start)
%F A154635 a(n) = (((4-sqrt(10))^n*(-8+sqrt(10))+(4+sqrt(10))^n*(8+sqrt(10))))/(12*sqrt(10)) for n>0.
%F A154635 a(n) = 8*a(n-1) - 6*a(n-2) for n>2.
%F A154635 (End)
%e A154635 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.
%t A154635 CoefficientList[Series[(1 - z) (1 - 5 z)/(1 - 8 z + 6 z^2), {z, 0, 100}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jul 03 2011 *)
%o A154635 (PARI) Vec((1-x)*(1-5*x)/(1-8*x+6*x^2) + O(x^30)) \\ _Colin Barker_, Nov 16 2016
%Y A154635 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.
%K A154635 easy,nonn
%O A154635 0,2
%A A154635 _Colin Mallows_, Jan 13 2009