Ehrhart polynomial for lattice squares, cubes and hypercubes

In this paper we are constructing integer lattice squares, cubes or hypercubes in $\mathbb R^d$ with $d\in \{2,3,4\}$. For squares and cubes we find a complete description of their Ehrhart polynomial. For hypercubes, we compute one of the coefficients and show that there exists a simple linear relation between the other two unknown coefficients. This allows as to formulate a conjecture of what the Ehrhart polynomial is in this case.