Applications of a determinant F-L identity (Q2883419)

The sequence \(\{w_n\}\) defined by NEWLINE\[NEWLINEw_{n+k} = a_1w_{n+k-1} + ... + a_{k-1}w_{n+1} + a_kw_n\tag{*} NEWLINE\]NEWLINE with initial conditions NEWLINE\[NEWLINEw_0 = c_0, w_1 = c_1, ..., w_{k-1} = c_{k-1},NEWLINE\]NEWLINE where \(a_1, ..., a_k\) and \(c_0, ..., c_{k-1}\) are complex constants, is called \(k\)-th order Fibonacci-Lucas sequence, or simply, an F-L sequence. Its characteristic polynomial is defined by NEWLINE\[NEWLINEf(x) = x^k - a_1x^{k-1} - ... - a_{k-1} x - a_k.NEWLINE\]NEWLINE Let \(\Omega(a_1, ..., a_k)\) be the set of all F-L sequences, satisfying (*). Some interesting properties of the elements of set \(\Omega(a_1,..., a_k)\) are discussed.