On Points Whose Coordinates Are Terms of a Linear Recurrence*

Let R = {Rn}TM be a second-order recurrent sequence (generalized Fibonacci sequence) of integers defined by Rn = ARn_l-BRn_2 (forn>l), where the initial terms are R$ = 0, Rx = 1, and A and B are fixed nonzero integers. Let a and j8 be the roots of the characteristic polynomial x Ax + B. We will assume that the discriminant D = A -AB > 0 and D is not a perfect square. From this, it follows that the sequence R is not degenerate, i.e., a I j3 is not a root of unity. In this case, a and j3 are two irrational real numbers and |a |^ | j3| , so we can suppose that |a|>|]3|. Also, 0<j3iffO<v4-i?. And 0< j3<l holds iff 0<B(A-B-l). It is well known that the terms of R can be given by