[seqfan] Re: Integral Apollonian circle packings

[Rick Shepherd]

Thanks to all for the references and the answer!

Although my reading about and understanding of this topic is still limited,
I believe it may yet be of some interest to explore with sequences where
(i.e., at what generation) these duplicate curvatures appear.  (I don't
have time to do this now.)

An easy observation:
By the definitions of integral Apollonian circle packing (or Apollonian
gasket) and of curvature (or bend), then each such packing of an outer
circle with curvature -n gives an infinite sum of reciprocals of squares
totaling 1/n^2.  For example, 1 = sum_{n=2..infty} A042946(n)/A042944(n)^2
(assuming indices matched as they should but don't currently) as these
sequences describe a unit circle packing.  Since Apollonian circle packings
could be nested (to arbitrary depths), an infinite number of such infinite
sums could potentially be derived for each total -- while ultimately also
making the largest (duplicate) circles in this type of packing as small as
desired.

Rick