If 
is a recursive sequence, then the set of all
such that 
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 
with 
.
Skolem-Mahler-Lech Theorem
Explore with Wolfram|Alpha
References
Lech, C. "A Note on Recurring Series." Ark. Mat. 2, 417-421, 1953.Myerson, G. and van der Poorten, A. J. "Some Problems Concerning Recurrence Sequences." Amer. Math. Monthly 102, 698-705, 1995.Referenced on Wolfram|Alpha
Skolem-Mahler-Lech TheoremCite this as:
Weisstein, Eric W. "Skolem-Mahler-Lech Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Skolem-Mahler-LechTheorem.html