Reduced diophantine quadruples with the binary recurrence G_n = AG_n–1 – G_n–2

Given a positive integer A ≠ 2. In this paper, we show that there do not exist two positive integer pairs {a,b} ≠ {c,d} such that the values of ac+1, ad+1 and bc+1, bd+1 are the terms of the sequence {Gn}_n≥0 which satisfies the recurrence relation G_n = AG_n-1 - G_n-2 with the initial values G₀ = 0, G₁ = 1.