A new factorization of the generalized period-doubling sequences through kernel words and gaps sequences

TL;DR

This paper introduces a novel factorization method for generalized period-doubling sequences over multiple alphabets, utilizing kernel words and gap sequences to analyze their combinatorial and arithmetic properties.

Contribution

It presents a new factorization approach for period-doubling sequences using kernel words and gap sequences, extending to sequences over larger alphabets.

Findings

Defined kernel words for period-doubling sequences

Established relationships between gap sequences and kernel words

Provided a generalized factorization method for k-letter alphabets

Abstract

In this paper, we study some new factorizations of period-doubling sequences over a $k$-letter alphabet, where $k\geq 2$. First, we define the combinatorial and arithmetic properties of these sequences. Then, we define the kernel words of period-doubling sequences and demonstrate how to factorize a binary sequence using its kernel words. Next, we define gap sequences for period-doubling sequences and explore their relationship with kernel words. Lastly, we present a factorization of period-doubling sequences for $k\geq 3$ based on kernel words and gap sequences.