Period patterns of certain \(k\)th-order linear recurrences over a finite field (Q2883409)

Let \(\mathbb F_q\) denote the finite field with \(q\) elements. Let \(u(a_1, \dots, a_k)\) denote the \(k\)th-order unit sequence satisfying the \(k\)th-order recursion relation NEWLINE\[NEWLINEu_{n+k} = a_1 u_{n+k-1} - a_2 u_{n+k-2} + \dots + (-1)^{k+1} a_k u_nNEWLINE\]NEWLINE with parameters \(a_1, \dots, a_k\) in \(\mathbb F_q\) and initial terms \(u_0 = u_1 = \dots = u_{k-2} = 0\), \(u_{k-1} = 1.\) Associated with \(u(a_1, \dots, a_k)\) is the characteristic polynomial NEWLINE\[NEWLINEf(x) = x^k - a_1 x^{k-1} + a_2 x^{k-2} + \dots + (-1)^{k} a_k.NEWLINE\]NEWLINE Continuing previous author's research, in this paper it is shown that for certain fixed \(k\)th parameters \(a_k\) and \(a'_k\), the number of unit sequences \(u(a_1, \dots, a_k)\) having a given period indeed equals the number of unit sequences \(u(a'_1, \dots, a'_k)\) having the same period, where \((a_1, \dots, a_{k-1})\) and \((a'_1, \dots, a'_{k-1})\) both vary over all ordered \((k-1)\)-tuples in \(\mathbb F_q\).