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%I #6 Aug 18 2017 16:05:47
%S 1,3,20,174,1170,8454
%N a(n) is the ratio of the sum of the bends of the spheres that are added in the n-th generation of Apollonian packing of three-dimensional spheres, using "strategy (b)" to count them (see the reference), to the sum of the bends of the initial five mutually tangent spheres.
%C In strategy (b) we count all spheres that can be generated (by reflection) from all quintuples that appeared in the previous generation.
%H C. L. Mallows, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Mallows/mallows8.html">Growing Apollonian Packings</a>, J. Integer Sequences, 12 (2009), article 09.2.1.
%e Starting with five spheres with bends 0,0,1,1,1, the first derived generation has 5 spheres with bends 1,1,1,3,3, so a(2) = 9/3 = 3.
%Y For other sequences relating to the 3-dimensional case, see A154638-A154645.
%K hard,more,nonn
%O 0,2
%A _Colin Mallows_, Jan 13 2009
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