On convergence properties of Fibonacci-like conditional sequences (Q2786864)

Fibonacci-like conditional sequence is defined in [\textit{D. Panario} et al., Integers 13, Paper A78, 14 p. (2013; Zbl 1329.11017)] for \(n \geq 2\) by NEWLINE\[NEWLINEv_n=\begin{cases} a_0v_{n-1}+b_0v_{n-2}, & \text{ if } n \equiv 0\pmod{k} \\ a_1v_{n-1}+b_1v_{n-2}, & \text{ if } n \equiv 1 \pmod{k} \\ & \vdots \\ a_{k-1}v_{n-1}+b_{k-1}v_{n-2}, & \text{ if } n \equiv k - 1\pmod {k} \end{cases}NEWLINE\]NEWLINE with arbitrary initial conditions \(v_0, v_1\) and \(a_0, a_1,\dots,a_{k-1}, b_0, b_1,\dots, b_{k-1}\) being non-zero numbers.NEWLINENEWLINEIn the paper is shown the boundary, to which the sequence \(\left \{\frac{v_{nk+r}}{v_{nk+r-1}} \right \}\) converges for \(n\to\infty\).