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

If P⊂Rd is a rational polytope, then iP(n)≔#(nP∩Zd) is a quasi-polynomial in n, called the Ehrhart quasi-polynomial of P. The minimum period of iP(n) must divide D(P)=min{n∈Z>0:nP is an integral polytope}. Few examples are known where the minimum period is not exactly D(P). We show that for any D, there is a 2-dimensional triangle P such that D(P)=D 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.