Counting Determinants of Fibonacci-hessenberg Matrices Using Lu Factorizations

is a Hessenberg matrix and its determinant is F2n+2. Furthermore, a Hessenberg matrix is said to be a Fibonacci-Hessenberg matrix [2] if its determinant is in the form tFn−1 + Fn−2 or Fn−1 + tFn−2 for some real or complex number t. In [1] several types of Hessenberg matrices whose determinants are Fibonacci numbers were calculated by using the basic definition of the determinant as a signed sum over the symmetric group. In this paper, we count the determinants of certain Hessenberg matrices by firstly investigating the feasibility of LU factorizations, i.e., a lower triangular matrix with unit main diagonal and an upper triangular matrix. Furthermore, the factorization is unique. As we know,the determinant of a triangular matrix is the product of its main diagonal entries. Then we can calculate easily the determinant of a given Hessenberg matrix by multiplying the diagonal entries of the corresponding upper triangular