On A Family Of 2-Automatic Sequences Derived From Ultimately Periodic Sequences And Generating Algebraic Continued Fractions In F2((1/T))

TL;DR

This paper introduces a family of 2-automatic sequences derived from ultimately periodic sequences, which generate algebraic continued fractions over F_2((1/T)), expanding understanding of their structure and properties.

Contribution

It describes a new family of sequences linked to algebraic continued fractions in F_2((1/T)), generalizing previous specific examples and connecting automatic sequences with algebraic continued fractions.

Findings

Includes a classical example previously studied

Establishes a link between 2-automatic sequences and algebraic continued fractions

Provides a framework for generating such sequences from ultimately periodic sequences

Abstract

By replacing the letters to polynomials in F_2[t], an infinite word, over a finite alphabet, can be seen as the sequence of partial quotients of a continued fraction in F_2((1/t)). Here is described a family of such infinite words, corresponding to continued fractions which are algebraic over F_2(t). This family includes a classical example already studied in different previous works by Y. Hu, G-N. Han, Y. Bugeaud and the author of this note.