lean_workbook_plus_22862

Let $k$ be a positive integer. The sequence $(a_n)_{n \geq 0}$ is defined by $a_0=0$ and $a_{n+1}=(a_n+1)k+(k+1)a_n+2\sqrt{k(k+1)a_n(a_n+1)}, \qquad n=0,1, \ldots.$ Prove that all $a_n$ are positive integers.