When a constant subsequence implies ultimate periodicity
Volume 67 / 2019
Bulletin Polish Acad. Sci. Math. 67 (2019), 41-51
MSC: 11B50, 11B83.
DOI: 10.4064/ba8174-4-2019
Published online: 6 May 2019
Abstract
We show a curious property of sequences given by the recurrence $a_0 = h_1(0)$, $a_n = f(n)a_{n-1} + h_1(n)h_2(n)^n$, $n \gt 0$, where $f,h_1,h_2 \in \mathbb {Z}[X]$. Namely, if the sequence $(a_{kn+l})_{n\in \mathbb {N}}$ is constant for some $k\in \mathbb {N}_+$ and $l\in \mathbb {N}$, then either $(a_{2n+1})_{n\in \mathbb {N}}=(0)_{n\in \mathbb {N}}$ and $(a_{2n})_{n\in \mathbb {N}}$ is a geometric progression, or $(a_{n})_{n\in \mathbb {N}}$ is ultimately periodic with period dividing $2$.
