Fix an integer $d\geq 1$, and let $n\geq 1$. Drawing hyperplanes between all the $d$-sets of lattice points on the boundary of the hypercube $[0,n]^d\subseteq \mathbf{R}^d$ defines a partition of $[0,n]^d$ into several distinct polytopes; let $a(n,d)$ denote the number of such polytopes. (Note that $a(n,d)$ is divisible by $2^d$.) For instance, $a(1,2) = 4$ and $a(2,2) = 56$. What can be said about the sequence $a(n,d)$ as $n$ and $d$ vary? (For example, is there a nice form for the generating function?)