lean_workbook_plus_15084

Let $\mathbb Z^+$ denote the set of all nonnegative integers. Define a recurrent sequence $\left(a_n\right)_{n\in\mathbb{Z}^+}$ by $a_0=a_1=1$ and the recurrence equation $a_{n+1}a_{n-1}=a_n\left(a_n+1\right)$ for every positive integer n. Prove that $a_n$ is an integer for every $n\in\mathbb{Z}^+$.