Anti-recurrence sequences

TL;DR

This paper investigates anti-recurrence sequences, proposing a conjecture that they are sums of linear progressions and automatic sequences, and provides a solution under certain restrictions.

Contribution

It proves the meta-conjecture for anti-recurrence sequences under specific linear form restrictions, advancing understanding of their structure.

Findings

Confirmed the meta-conjecture for restricted cases

Demonstrated that anti-recurrence sequences can be decomposed into linear and automatic parts

Extended previous theoretical frameworks on anti-recurrence sequences

Abstract

We extend previous work on anti-recurrence sequences of Kimberling and Moses, Zaslavsky, and Bosma et al. Kimberling and Moses have formulated several questions on these sequences, which can be combined into the meta-conjecture that anti-recurrence sequences are sums of linear progressions and automatic sequences. We solve this conjecture under a restriction on the linear form that generates the anti-recurrence.