  Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart
states that the number of lattice points in the dilations $P(n) = nP$ is a
quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of
P(n) to be arbitrary rational functions in $n$. In this case we prove that the
number of lattice points in P(n) is a quasi-polynomial for $n$ sufficiently
large. Our work was motivated by a conjecture of Ehrhart on the number of
solutions to parametrized linear Diophantine equations whose coefficients are
polynomials in $n$, and we explain how these two problems are related.
