Define the sequence $(a_k)_{k \ge 0}$ by $a_0 = 1$, $a_1 = 2$ and $a_{k+2} a_k = a_{k+1}^2 + 1$, $k = 0, 1, 2, \dots$, and check inductively that the $a_k$ are all positive integer numbers, the $n_k = a_{k+1} - a_k$ form a strictly increasing sequence of positive integer numbers, and $a_k$ and $a_{k+1}$ both divide $n_k^2 + 1$.