When a constant subsequence implies ultimate periodicity

Piotr Miska ¹

¹ Institute of Mathematics Faculty of Mathematics and Computer Science Jagiellonian University in Kraków Łojasiewicza 6 30-348 Kraków, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 67 (2019) no. 1, pp. 41-51

Résumé

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 $2$.

Détail

DOI : 10.4064/ba8174-4-2019

Keywords: curious property sequences given recurrence n where mathbb namely sequence mathbb constant mathbb mathbb either mathbb mathbb mathbb geometric progression mathbb ultimately periodic period dividing nbsp