  Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to
K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$
for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in
$K[z]$. Given that $P_d(z)$ is a nonzero constant polynomial, we show that the
set of $n\in \mathbb{N}$ for which $f(n)=0$ is a union of finitely many
arithmetic progressions and a finite set. This generalizes the
Skolem-Mahler-Lech theorem, which assumes that $f(n)$ satisfies a linear
recurrence. We discuss examples and connections to the set of zero coefficients
of a power series satisfying a homogeneous linear differential equation with
rational function coefficients.
