IMO 1985 LL CAN14

Let k be a positive integer. Define u0 = 0, u1 = 1, and

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IMO 1985 LL CAN14

Origin: CAN

Problem

Let k be a positive integer. Define u0 = 0, u1 = 1, and un = kun−1 −un−2, n \geq2. Show that for each integer n, the number u3 1 + u3 2 + \cdot \cdot \cdot + u3 n is a multiple of u1 + u2 + \cdot \cdot \cdot + un.