Ehrhart's polynomial for equilateral triangles in \(\mathbb Z^3\) (Q2869303)

To a \(d\)-dimensional lattice polytope \(P \subset\mathbb R^d\) (i.e., the convex hull of finitely many points in in \(\mathbb Z^d\)), we associate the integer-point counting function \(L_P(t) := \# \left( tP \cap {\mathbb Z}^d \right)\), defined for positive integers \(t\). Ehrhart's famous theorem [\textit{E. Ehrhart}, C. R. Acad. Sci., Paris 254, 616--618 (1962; Zbl 0100.27601)] asserts that \(L_P(t)\) is a polynomial in \(t\).NEWLINENEWLINEBuilding on previous work by the same author, the paper under review gives a formula for the Ehrhart polynomial of an equilateral lattice triangle in \(\mathbb R^3\). This allows for a classification of sorts for Ehrhart polynomials of equilateral triangles; e.g., the paper shows that for any prime number \(p\equiv 1\) or \(3 \bmod 8\), there exists an equilateral lattice triangle \(P\) with Ehrhart polynomial \(L_P(t) = {1 \over 2} (pt+2) (t+1)\).