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Discrete Mathematics and Theoretical Computer Science (subm.), by the authors, 1--rev Sums of Digits, Overlaps, and Palindromes
 

Summary: Discrete Mathematics and Theoretical Computer Science (subm.), by the authors, 1--rev
Sums of Digits, Overlaps, and Palindromes
Jean­Paul Allouche 1 and Jeffrey Shallit 2y
1 CNRS, Laboratoire de Recherche en Informatique, B“atiment 490, F­91405 Orsay Cedex, France
2 Department of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
received 3 Aug 1999, revised 8 Feb 2000, accepted ???.
Let sk (n) denote the sum of the digits in the base­k representation of n. In a celebrated paper, Thue
showed that the infinite word (s2(n) mod 2)n–0 is overlap­free, i.e., contains no subword of the form
axaxa, where x is any finite word and a is a single symbol. Let k; m be integers with k – 2, m – 1.
In this paper, generalizing Thue's result, we prove that the infinite word tk;m := (sk (n) mod m)n–0 is
overlap­free if and only if m – k. We also prove that tk;m contains arbitrarily long squares (i.e., subwords
of the form xx where x is nonempty), and contains arbitrarily long palindromes if and only if m ź 2.
Keywords: sum of digits, overlap­free sequence, palindrome
Contents
1 Introduction 1
2 Some useful lemmas 2
3 Proof of the main theorem 4
4 Squares in the sequence t k;m 8
5 Palindromes in t k;m 9
1 Introduction

  

Source: Allouche, Jean-Paul - Laboratoire de Recherche en Informatique, Université de Paris-Sud 11

 

Collections: Computer Technologies and Information Sciences