Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #6 Aug 18 2017 16:01:52
%S 1,3,20,108,630,3570,20460
%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 (a)" to count them (see the reference), to the sum of the bends of the initial five mutually tangent spheres.
%C In strategy (a) we do not count spheres that can be obtained (by reflection) from the "extra" quintuples of spheres that appear 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