Runs, Squares, Palindromes, and Unbordered Factors of a Family of Binary Pattern Sequences with the All-One Pattern

TL;DR

This paper investigates the structural properties of binary pattern sequences with all-one patterns, analyzing runs, squares, palindromes, and unbordered factors, and employs computational tools and number theory for proofs.

Contribution

It introduces a detailed classification of maximal runs, squares, and palindromes in these sequences, and links their properties to well-known numerical sequences.

Findings

Classification of maximal run lengths into five categories.

Description of palindrome lengths including locally maximal and second to fifth-largest.

Identification of unbordered factors at lengths that are powers of two.

Abstract

This paper presents results on maximal runs, order of squares, palindromes, and unbordered factors of members of the family of binary pattern sequences with the all-one pattern. Restricting ourselves to binary pattern sequences with the all-one pattern with at least three ones, five categories of maximal run lengths and 3 categories of orders of squares are presented, palindromes with locally maximal length as well as palindromes with the second to fifth-largest palindrome lengths are described, and unbordered factors of lengths powers of two are presented. Interestingly, the characteristic functions of specified prefixes of sequences of the 2-kernel of these sequences can be formulated using the Vile and Jacobsthal sequences. Both Mathematica and Walnut are employed for exploratory pattern analysis. Proofs are based on a correspondence between binary strings under concatenation and integers under addition and multiplication. It is observed that this correspondence seems most efficacious for proofs of theorems whose statements are classified at low levels in the arithmetic hierarchy.