BALTIC-WAY 2014 Problem 20

Consider a sequence of positive integers $a_{1}, a_{2}, a_{3}, \ldots$ such that for $k \geqslant 2$ we have

$$a_{k+1}=\frac{a_{k}+a_{k-1}}{2015^{i}}$$

where $2015^{i}$ is the maximal power of 2015 that divides $a_{k}+a_{k-1}$. Prove that if this sequence is periodic then its period is divisible by 3 .