A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers q Ays ße Nalli * , Haci Civciv Department of Mathematics, Faculty of Art and Science, Selcuk University, 42031 Konya, Turkey Accepted 23 July 2007 Communicated by Dr. Paolo Grigolini Abstract In this paper, we construct the symmetric tridiagonal family of matrices M a;b ðkÞ; k ¼ 1; 2; ... whose determinants form any linear subsequence of the Fibonacci numbers. Furthermore, we construct the symmetric tridiagonal family of matrices T a;b ðkÞ; k ¼ 1; 2; ... whose determinants form any linear subsequence of the Lucas numbers. Thus we give a generalization of the presented in Cahill and Narayan (2004) [Cahill ND, Narayan DA. Fibonacci and Lucas numbers as tridiagonal matrix determinants. Fibonacci Quart 2004;42(3):216–21]. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction There are many known connections between determinants of tridiagonal matrices and the Fibonacci and Lucas numbers. For example, Strang [5,6] presents a family of tridiagonal matrices given by: M ðnÞ¼ 3 1 1 3 1 1 : : : : : : : : : 3 1 1 3 0 B B B B B B B B B B B @ 1 C C C C C C C C C C C A ; ð1:1Þ where M(n) is n · n . It is easy to show by induction that the determinants jM(k)j are the Fibonacci numbers F 2k+2 . Another example is the family of tridiagonal matrices given by: 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.07.069 q This paper is in final form and no version of it will be submitted for publication elsewhere. * Corresponding author. E-mail addresses: aysenalli@gmail.com (A. Nalli), hacicivciv@gmail.com (H. Civciv). Chaos, Solitons and Fractals 40 (2009) 355–361 www.elsevier.com/locate/chaos