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%I #6 Aug 18 2017 16:07:26
%S 5,5,20,90,330,1290
%N a(n) is the number of spheres that are added in the n-th generation of Apollonian packing of three-dimensional spheres, starting with five mutually tangent spheres and using "strategy (b)" to count them (see the reference).
%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 For a(3), we apply reflection to the 20 quintuples that were generated in the second generation and also to the 10 "extra" quintuples which appear at this stage, each consisting of two spheres in the initial configuration and three from generation 2. These quintuples will appear as ABA = BAB in the third generation.
%Y For other sequences relating to the 3-dimensional case, see A154638-A154645.
%K hard,more,nonn
%O 0,1
%A _Colin Mallows_, Jan 13 2009
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