Two-periodic ternary recurrences and their Binet-formula (Q2913250)

The two-periodic ternary recurrence sequence is defined by relations \(\gamma _n=a\gamma _{n-1}+b\gamma _{n-2}+c\gamma _{n-3}\) if \(n\) is even and \(\gamma _n=d\gamma _{n-1}+e\gamma _{n-2}+f\gamma _{n-3}\) if \(n\) is odd. In this paper, Cooper's approach [\textit{C. Cooper}, Congr. Numerantium 200, 95--106 (2010; Zbl 1204.11023)] is applied to obtain the recurrence relation \(\gamma _n=(ad+b+e)\gamma _{n-2}+(af-be+cd)\gamma _{n-4}+cf\gamma _{n-6}\) of order six for \(\gamma _n\) and then the Binet-formulae (Theorems 2-4) are derived using the fundamental theorem of linear recurrences. The three resulting cases correspond to the number of distinct zeros of the characteristic polynomial of the recurrence.