Skolem-Mahler-Lerch Theorem

If $\{a_0, a_1, \ldots\}$ is a Recurrence Sequence, then the set of all $k$ such that $a_k=0$ is the union of a finite (possibly Empty) set and a finite number (possibly zero) of full arithmetical progressions, where a full arithmetic progression is a set of the form $\{r, r+d, r+2d, \ldots\}$ with $r\in[0,d).$

References

Myerson, G. and van der Poorten, A. J. ``Some Problems Concerning Recurrence Sequences.'' Amer. Math. Monthly 102, 698-705, 1995.