There have been a number of articles on the relation between the terms of the Fibonacci and Lucas sequences and how they are closely related to trigonometric and hyperbolic functions and their properties [1]. This article is based on other integer sequences. It sets out to determine other pairs of such sequences that have the same relation as the Fibonacci and Lucas have to each other. So we shall be concerned with second order recurrence relations with constant coefficients:u_n= au_(n-1) + bu _(n-2) (α, b independent of n) and pairs of sequences (u-n) and (v_n) that each satisfy it. We seek a condition that ensures the pair of sequences behave as the Fibonacci-Lucas pair behave. The tools used are entirely elementary, so in particular, no generating functions are employed. It is hoped that this makes the article suitable for students familiar with recurrences and some basic algebra. The ubiquitous Fibonacci sequence (F_n) needs little introduction [2], but its partner the Lucas sequence (L_n) is much less well known. They obey the same recurrence relation u_n = u_n-1 +u_n-2 (1) (so a = b = 1)but have different starting values The Fibonacci sequence begins, F_0 = 0, F_1 =1 and note that we adopt the convention that the first term is F_0(= 0).The Luces sequence has two different starting values, L_0 = 2,L_1 = 1. Note that two inital values and a second order recurrence|(with constant coefficients) are sufficent to define a unique sequence.
展开▼
