A problem I am working on involves the number, $A_h^n$, of points $\mathbf{q} = (q_1, \ldots, q_n)$ in the integer lattice $\Bbb{Z}^n \subseteq \Bbb{R}^n$ such that $\|\mathbf{q}\|_{\infty} = \max_{i = 1}^n|q_i| = h$, for given $, n, h \in \Bbb{N}^+$. I find that: $$ A_h^n = 2^n \sum_{i=1}^n (h-1)^{i-1}h^{n-i} $$ (where summand $i$ gives the number of $\mathbf{q}$ where the $q_j$ are all positive and $i$ is the least index such that $q_i = h$). I don't think I need it for the problem in hand, but I am curious to know if there is a nice closed form for $A_h^n$.
[Note: I goofed. See Robert's answer.]
Your formula is not quite right. For one thing, your formula would always give an even number, but it should be odd. Note that $A_h^n = F_h^n - F_{h-1}^n$, where $F_h^n$ is the number of points $q$ with $\|q\|_\infty \le h$, and this is easy to calculate.