Acta Math. Univ. Comenianae
First, we calculate the Ehrhart polynomial associated with an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedra and those for regular lattice octahedra. These relations allow one to reduce the calculation of these polynomials to only one coefficient.
Ionascu, Eugen J., "Lattice platonic solids and their Ehrhart polynomial" (2013). Faculty Bibliography. 678.