A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers

In this paper, we construct the symmetric tridiagonal family of matrices M-α,-β(k),k=1,2,... whose determinants form any linear subsequence of the Fibonacci numbers. Furthermore, we construct the symmetric tridiagonal family of matrices T-α,-β(k),k=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].