Possibly too easy for stack exchange, but...
Consider a cubic close packing, or face centred cubic, arrangement of balls or radius $1$ in dimension $3$.
Suppose that the origin is the centre of one of the balls. Let $r$ be the distance to the origin.
I wish to count how many balls are located at a given distance.
That is easy to do in dimension $2$. The densest close packing arrangement is an hexagonal lattice, therefore we can count balls placed on dilated hexagons: 6 when $r=2$ and in general $6n$ when $r=2n$.
I would like a similar estimate in dimension 3. At $r=2$, there are exactly $12$ balls. At $r=2n$, how many balls? $12n^2$?