Symmetric placing of balls Suppose I have a round domain and put my first ball in the very center (this ball alone constitutes the $0$th ring).  Now around this ring (this first ball) I want to add another ring of balls in a fashion so that the distance between any ball and any of its neighbors is equal.  Clearly the symmetric hexagon pattern will make the distance between all neighboring ball (including the inner "ring") equal distance. Now how to I add the second ring?  And in general, given a patter for ring $n$ how do I build ring $(n+1)$?

You can continue the hexagon pattern.  The first ring will have six balls, the second $12$, and the $n^{\text{th}}$ will have $6n$.  This gives the centered hexagonal numbers $3n(n+1)+1$  It keeps each ball the same distance from its neighbors.
