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Consider the sequences {u,} and {v"} generated by u,+1 = Pun qun-I and Vn+1 = Pvn qvn-I , n > 1 , where uo = 0, ul = 1, vo = 2, v, = p, with p and q real and nonzero. The Fibonacci sequence and the Lucas sequence are special cases of {un} and {vn}, respectively. Define rn = un+d/un, Rn = vn+d/vn, where d is a positive integer. McCabe and Phillips showed that for d = 1, applying one step of Aitken acceleration to any appropriate triple of elements of {rn} yields another element of {rn}. They also proved for d = 1 that if a step of the Newton-Raphson method or the secant method is applied to elements of {rn } in solving the characteristic equation x2 px + q = 0, then the result is an element of {rn }. The above results are obtained for d > 1. It is shown that if any of the above methods is applied to elements of {Rn}, then the result is an element of {rn}. The application of certain higher-order iterative procedures, such as Halley's method, to elements of {rn } and {Rn } is also investigated. Fibonacci and Lucas numbers appear repeatedly in the works of the father of computational number theory, D. H. Lehmer, who contributed also to numerical analysis, notably [5]. To his memory is dedicated this extension of results of McCabe and Phillips [6] and Jamieson [4] about applying iterative formulas for solving nonlinear equations to ratios of generalized Fibonacci numbers.

Generalized Fibonacci and Lucas sequences and rootfinding methods