Abstract
We shall consider the general sequence satisfying the recurrence relation which generates the Fibonacci and Lucas sequences {Fn} and {Ln}
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
And
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}
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References
Hoggatt, Verner E. Jr. Fibonacci and Lucas Numbers. Houghton Mifflin, Boston, 1969.
Long, Calvin T., “Discovering Fibonacci Identities,” The Fibonacci Quarterly 24 (1986): pp. 160–167.
Vajda, S. Fibonacci & Lucas Numbers, and the Golden Section. John Wiley & Sons, New York, 1989.
Vorob’ev, N. N. Fibonacci Numbers. Pergamon, 1961.
Walton, J. E. and Horadam, A. F. “Some Aspects of Generalized Fibonacci Numbers,” The Fibonacci Quarterly 12 (1974): pp. 241–250.
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© 1991 Springer Science+Business Media Dordrecht
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Freitag, H.T., Phillips, G.M. (1991). On Co-Related Sequences Involving Generalized Fibonacci Numbers. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3586-3_14
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DOI: https://doi.org/10.1007/978-94-011-3586-3_14
Publisher Name: Springer, Dordrecht
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