The minimum period of the Ehrhart quasi-polynomial of a rational polytope

If is a rational polytope, then is a quasi-polynomial in n, called the Ehrhart quasi-polynomial of P. The minimum period of iP(n) must divide . Few examples are known where the minimum period is not exactly . We show that for any , there is a 2-dimensional triangle P such that but such that the minimum period of iP(n) is 1, that is, iP(n) is a polynomial in n. We also characterize all polygons P such that iP(n) is a polynomial. In addition, we provide a counterexample to a conjecture by T. Zaslavsky about the periods of the coefficients of the Ehrhart quasi-polynomial.