GENERALIZED LUCAS NUMBERS OF THE FORM 5kx(2) AND 7kx(2)

Abstract

Generalized Fibonacci and Lucas sequences (U-n) and (V-n) are defined by the recurrence relations Un+1 = PUn +QU(n-1) and Vn+1 = PVn + QV(n-1), n >= 1, with initial conditions U-0 = 0, U-1 = 1 and V-0 = 2, V-1 = P. This paper deals with Fibonacci and Lucas numbers of the form U-n (P, Q) and V-n (P, Q) with the special consideration that P >= 3 is odd and Q = -1. Under these consideration, we solve the equations V-n = 5kx(2), V-n = 7kx(2), V-n = 5kx(2)+/- 1, and V-n = 7kx(2)+/- 1 when k vertical bar P with k > 1. Moreover, we solve the equations V-n = 5x(2)+/- 1 and V-n = 7x(2 +/-)1.