Discrete Mathematics & Theoretical Computer Science, Vol 4, No 1 (2000)

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DMTCS vol 4 no 1 (2000), pp. 1-10

Discrete Mathematics & Theoretical Computer Science


Volume 4 n° 1 (2000), pp. 1-10

author:Jean-Paul Allouche and Jeffrey Shallit
title:Sums of Digits, Overlaps, and Palindromes
keywords:sum of digits, overlap-free sequence, palindrome
abstract:Let s_k(n) denote the sum of the digits in the base-k representation of n. In a celebrated paper, Thue showed that the infinite word s_2(n) mod 2)_{n>=0} is overlap-free, 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 t_{k,m} := (s_k(n) mod m)_{n>=0} is overlap-free if and only if m>=k. We also prove that t_{k,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<= 2.
reference: Jean-Paul Allouche and Jeffrey Shallit (2000), Sums of Digits, Overlaps, and Palindromes, Discrete Mathematics and Theoretical Computer Science 4, pp. 1-10
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