On The Second Order Linear Recurrences By Tridiagonal Matrices

In this paper, we consider the relationships between the second order linear recurrences and the permanents and determinants of tridiagonal matrices. 1. Introduction The well-known Fibonacci, Lucas and Pell numbers can be generalized as follows: Let A and B be nonzero, relatively prime integers such that D = A 4B 6= 0: De…ne sequences fung and fvng by, for all n 2 (see [10]), un = Aun 1 Bun 2 (1.1) vn = Avn 1 Bvn 2 (1.2) where u0 = 0; u1 = 1 and v0 = 2; v1 = A: If A = 1 and B = 1; then un = Fn (the nth Fibonacci number) and vn = Ln (the nth Lucas number). If A = 2 and B = 1; then un = Pn ( the nth Pell number). An alternative is to let the roots of the equation t At+B = 0 be, for n 0 un = n n and vn = n + : (1.3) Also it is well-known that + = A and = B: The permanent of an n-square matrix A = (aij) is de…ned by