A diluted version of the problem of the existence of the Hofstadter sequence
Number Theory
2023-11-27 v1 Dynamical Systems
Abstract
We investigate the conditions on an integer sequence f(n), n 2 N, with f(1) = 0, such that the sequence q(n), computed recursively via q(n) = q(n - q(n - 1)) + f(n), with q(1) = 1, exists. We prove that f(n + 1) - f(n) in {0,1}, n > 0, is a sufficient but not necessary condition for the existence of sequence q. Sequences q defined in this way typically display non-trivial dynamics: in particular, they are generally aperiodic with no obvious patterns. We discuss and illustrate this behaviour with some examples.
Keywords
Cite
@article{arxiv.2311.13854,
title = {A diluted version of the problem of the existence of the Hofstadter sequence},
author = {Jonathan H. B. Deane and Guido Gentile},
journal= {arXiv preprint arXiv:2311.13854},
year = {2023}
}
Comments
17 pages, 4 figures