RIOPLATENSE 2019 Problem 6

Let $\alpha>1$ be a real number such that the sequence $a_n=\alpha\lfloor \alpha^n\rfloor- \lfloor \alpha^{n+1}\rfloor$ , with $n\geq 1$ , is periodic, that is, there is a positive integer $p$ such that $a_{n+p}=a_n$ for all $n$ . Prove that $\alpha$ is an integer.