Ehrhart's polynomial for equilateral triangles in ℤ3

In this paper we calculate the Ehrhart’s polynomial associated with a 2-dimensional regular polytope (i.e. equilateral triangles) in Z. The polynomial takes a relatively simple form in terms of the coordinates of the vertices of the triangle. We give some equivalent formula in terms of a parametrization of these objects which allows one to construct equilateral triangles with given properties. In particular, we show that given a prime number p which is equal to 1 or −5 (mod 8), there exists an equilateral triangle with integer coordinates whose Ehrhart polynomial is L(t) = (pt + 2)(t + 1)/2, t ∈ N.