Members of Binary Recurrences on Lines of the Pascal Triangle

In this paper, we look at a diophantine equation of the form un = ( x y ) , where (un)n≥0 is a binary recurrent sequence of integers. We show that if the pair of integers (x, y) belongs to a fixed line of the Pascal triangle, then the above equation has only finitely many positive integer solutions (n, x, y). A binary recurrent sequence of integers (un)n≥0 has u0, u1 ∈ Z and satisfies the recurrence un+2 = run+1 + sun (1) for all n ≥ 0, where r and s are some fixed nonzero integers such that the quadratic equation λ − rλ− s = 0 has two distinct nonzero roots α and β. In this case, the formula