  We consider the integer point transform $\sigma _P (\mathbf{x}) = \sum
_{\mathbf{m} \in P\cap \mathbb{Z}^n} \mathbf{x}^\mathbf{m} \in \mathbb C
[x_1^{\pm 1},\ldots, x_n^{\pm 1}]$ of a polytope $P\subset \mathbb{R}^n$. We
show that if $P$ is a lattice polytope then for any polytope $Q$ the sequence
$\lbrace \sigma _{kP+Q}(\mathbf{x})\rbrace _{k\geq 0}$ satisfies a multivariate
linear recursion that only depends on the vertices of $P$. We recover Brion's
Theorem and by applying our results to Schur polynomials we disprove a
conjecture of Alexandersson (2014).
