On the representation of k-generalized Fibonacci and Lucas numbers Ahmet Ali O ¨ cal, Naim Tuglu * , Ercan Altinis ßik Mathematics, Gazi University, Teknikokullar, 06015 Ankara, Turkey Abstract In this paper we give some determinantal and permanental representations of k-gen- eralized Fibonacci and Lucas numbers. We obtain the BinetÕs formula for these sequences by using our representations. Ó 2005 Elsevier Inc. All rights reserved. Keywords: k-Generalized Fibonacci numbers; k-Generalized Lucas numbers; Hessenberg matrices 1. Introduction Fibonacci numbers, Lucas numbers and their generalization have many interesting properties and applications to almost every fields of science and art. For the beauty and rich applications of these numbers and their relatives one can see KoshyÕs book and the nature. Besides the usual Fibonacci and Lucas numbers many kinds of generaliza- tions of these numbers have been presented in the literature (e.g. see [15,7,5,10,11,13]). One of these generalizations was given by Miles in 1960. 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.12.009 * Corresponding author. E-mail addresses: aliocal@gazi.edu.tr (A.A. O ¨ cal), naimtuglu@gazi.edu.tr (N. Tuglu), ealtinisik@gazi.edu.tr (E. Altinis ßik). Applied Mathematics and Computation 170 (2005) 584–596 www.elsevier.com/locate/amc