  We prove that the sequence of the characters of the Kirillov-Reshetikhin (KR)
modules $W_{m}^{(a)}, m\in \mathbb{Z}_{m\geq 0}$ associated to a node $a$ of
the Dynkin diagram of a complex simple Lie algebra $\mathfrak{g}$ satisfies a
linear recurrence relation except for some cases in types $E_7$ and $E_8$. To
this end we use the $Q$-system and the existing lattice point summation formula
for the decomposition of KR modules, known as domino removal rules when
$\mathfrak{g}$ is of classical type. As an application, we show how to reduce
some unproven lattice point summation formulas in exceptional types to finite
problems in linear algebra and also give a new proof of them in type $G_2$,
which is the only completely proven case when KR modules have an irreducible
summand with multiplicity greater than 1. We also apply the recurrence to prove
that the function $\dim W_{m}^{(a)}$ is a quasipolynomial in $m$ and establish
its properties. We conjecture that there exists a rational polytope such that
its Ehrhart quasipolynomial in $m$ is $\dim W_{m}^{(a)}$ and the lattice points
of its $m$-th dilate carry the same crystal structure as the crystal associated
with $W_{m}^{(a)}$.
