%I
%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 nth generation of Apollonian packing of threedimensional 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 3dimensional case, see A154638A154645.
%K hard,more,nonn
%O 0,2
%A _Colin Mallows_, Jan 13 2009
