Question 164
Let $a_1$, $a_2$, $r$, and $s$ be positive integers with $r$ and $s$ odd. The sequence $a_1$, $a_2$, $a_3$, $\dots$ is defined by
$$
a_{n+2} = r a_{n+1} + s a_n
$$
for all $n \ge 1$. Determine the maximum possible number of integers $1 \le l \le 2025$ such that $a_l$ divides $a_{l+1}$, over all possible choices of $a_1$, $a_2$, $r$, and $s$.
$$
a_{n+2} = r a_{n+1} + s a_n
$$
for all $n \ge 1$. Determine the maximum possible number of integers $1 \le l \le 2025$ such that $a_l$ divides $a_{l+1}$, over all possible choices of $a_1$, $a_2$, $r$, and $s$.