When a constant subsequence implies ultimate periodicity
Bulletin of the Polish Academy of Sciences. Mathematics, Tome 67 (2019) no. 1, pp. 41-51
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
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 $2$.
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
Affiliations des auteurs :
Piotr Miska 1
@article{10_4064_ba8174_4_2019,
author = {Piotr Miska},
title = {When a constant subsequence implies ultimate periodicity},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
pages = {41--51},
publisher = {mathdoc},
volume = {67},
number = {1},
year = {2019},
doi = {10.4064/ba8174-4-2019},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/ba8174-4-2019/}
}
TY - JOUR AU - Piotr Miska TI - When a constant subsequence implies ultimate periodicity JO - Bulletin of the Polish Academy of Sciences. Mathematics PY - 2019 SP - 41 EP - 51 VL - 67 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4064/ba8174-4-2019/ DO - 10.4064/ba8174-4-2019 LA - en ID - 10_4064_ba8174_4_2019 ER -
%0 Journal Article %A Piotr Miska %T When a constant subsequence implies ultimate periodicity %J Bulletin of the Polish Academy of Sciences. Mathematics %D 2019 %P 41-51 %V 67 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4064/ba8174-4-2019/ %R 10.4064/ba8174-4-2019 %G en %F 10_4064_ba8174_4_2019
Piotr Miska. When a constant subsequence implies ultimate periodicity. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 67 (2019) no. 1, pp. 41-51. doi: 10.4064/ba8174-4-2019
Cité par Sources :