Question 123
Let $k$ be an integer and $k \gt 1$. Define a sequence $\{a_n\}$ as follows: $a_0 = 0$, $a_1 = 1$ and $a_{n+1} = k a_n + a_{n-1}$ for $n = 1, 2, \dots$
Determine, with proof, all possible $k$ for which there exist non-negative integers $\ell, m$ ($\ell \ne m$) and positive integers $p, q$ such that $a_\ell + k a_p = a_m + k a_q$.
Determine, with proof, all possible $k$ for which there exist non-negative integers $\ell, m$ ($\ell \ne m$) and positive integers $p, q$ such that $a_\ell + k a_p = a_m + k a_q$.