On Co-Related Sequences Involving Generalized Fibonacci Numbers

We shall consider the general sequence satisfying the recurrence relation which generates the Fibonacci and Lucas sequences {Fn} and {Ln} 1 $$ {A_{n + 1}} = {A_n} + {A_{n - 1}},{\kern 1pt} \;\;n \in \mathbb{Z} $$ with A0, Al ∈ R given.(See Walton and Horadam [5] for a lengthy list of references on this sequence.) There is a nearly symmetrical relation between {Fn} and {Ln} exhibited by the well known identities 2 $$ {F_{n + 1}} + {F_{n - 1}} = {L_n} $$ And 3 $$ {L_{n + 1}} + {L_{n - 1}} = 5{F_n} $$ On surveying the very large number of identities involving {Fn} and {Ln} (see, for example, Hoggatt [1], Long [2], Vajda [3], Vorob’ev [4]) we have observed that some are equivalent with respect to the identities (2) and (3). We will illustrate this in §2. This encourages us to seek other pairs of sequences satisfying (1) which are also inter-related in a symmetrical way, as in (2) and (3) for {Fn} and {Ln}