ILL 1985 Problem 14

Let $k$ be a positive integer. Define $u_0 = 0, u_1 = 1$ , and $u_n=ku_{n-1}-u_{n-2} , n \geq 2.$ Show that for each integer $n$ , the number $u_1^3 + u_2^3 +\cdots+ u_n^3$ is a multiple of $u_1 + u_2 +\cdots+ u_n.$