On generalized Stanley sequences

TL;DR

This paper investigates generalized versions of Stanley sequences, which are constructed via a greedy algorithm to avoid k-term arithmetic progressions, and presents new theoretical results on their properties.

Contribution

The paper introduces and analyzes various generalizations of Stanley sequences, providing new theoretical insights into their structure and behavior.

Findings

Established properties of generalized Stanley sequences

Proved results on their growth and structure

Extended classical results to broader classes of sequences

Abstract

Let $\mathbb{N}$ denote the set of all nonnegative integers. Let $k\ge 3$ be an integer and $A_{0} = \{a_{1}, \dots{}, a_{t}\}$ $(a_{1} < \ldots< a_{t})$ be a nonnegative set which does not contain an arithmetic progression of length $k$. We denote $A = \{a_{1}, a_{2}, \dots{}\}$ defined by the following greedy algorithm: if $l \ge t$ and $a_{1}, \dots{}, a_{l}$ have already been defined, then $a_{l+1}$ is the smallest integer $a > a_{l}$ such that $\{a_{1}, \dots{}, a_{l}\} \cup \{a\}$ also does not contain a $k$-term arithmetic progression. This sequence $A$ is called the Stanley sequence of order $k$ generated by $A_{0}$. In this paper, we prove some results about various generalizations of the Stanley sequence.