More Kolakoski Sequences

TL;DR

This paper reviews the properties of the Kolakoski sequence and explores its generalizations over different two-letter alphabets, focusing on cases with mixed parity of letters and analyzing various combinatorial properties.

Contribution

It provides a comprehensive review of known properties and investigates new generalizations of the Kolakoski sequence for alphabets with mixed parity, highlighting differences from well-known substitution sequences.

Findings

Analysis of word and letter frequencies in generalized sequences

Identification of palindromes and squares within these sequences

Insights into the complexity of sequences with mixed parity letters

Abstract

Our goal in this article is to review the known properties of the mysterious Kolakoski sequence and at the same time look at generalizations of it over arbitrary two letter alphabets. Our primary focus will here be the case where one of the letters is odd while the other is even, since in the other cases the sequences in question can be rewritten as (well-known) primitive substitution sequences. We will look at word and letter frequencies, squares, palindromes and complexity.