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In this study, we show that there is no positive integer triple {a, b, c} such that all of ab+1, ac+1 and bc+1 are in the sequence {un}n≥0 satisfies the recurrence un = Aun−1−un−2 with the initial values u0 = 0, u1 = 1. Further, we investigate the analogous question for the quadruples {a, b, c, d} with abc + 1 = ux, bcd + 1 = uy , cda + 1 = uz and dab+ 1 = ut, and deduce the non-existence of such quadruples.

Diophantine triples and reduced quadruples with the Lucas sequence of recurrence un=Aun-1-un-2