CWMO 2010 Problem 5

Let $k$ be an integer and $k > 1$ . Define a sequence $\{a_n\}$ as follows: $a_0 = 0$ , $a_1 = 1$ , and $a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$ .

Determine, with proof, all possible $k$ for which there exist non-negative integers $l,m (l \not= m)$ and positive integers $p,q$ such that $a_l + ka_p = a_m + ka_q$ .