Vol. 8, No. 4, 2019

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Generalized Beatty sequences and complementary triples

Jean-Paul Allouche and F. Michel Dekking

Vol. 8 (2019), No. 4, 325–341
Abstract

A generalized Beatty sequence is a sequence V defined by V (n) = pnα + qn + r, for n = 1,2,, where α is a real number, and p,q,r are integers. Such sequences occur, for instance, in homomorphic embeddings of Sturmian languages in the integers.

We consider the question of characterizing pairs of integer triples (p,q,r),(s,t,u) such that the two sequences V (n) = (pnα + qn + r) and W(n) = (snα + tn + u) are complementary (their image sets are disjoint and cover the positive integers). Most of our results are for the case that α is the golden mean, but we show how some of them generalize to arbitrary quadratic irrationals.

We also study triples of sequences V i = (pinα + qin + ri), i = 1,2,3 that are complementary in the same sense.

Keywords
generalized Beatty sequences, complementary pairs and triples, morphic words, return words, Kimberling transform
Mathematical Subject Classification 2010
Primary: 11B83, 11B85, 11D09, 11J70, 68R15
Milestones
Received: 28 December 2018
Revised: 22 May 2019
Accepted: 5 June 2019
Published: 11 October 2019
Authors
Jean-Paul Allouche
CNRS, IMJ-PRG, Sorbonne Université
Paris
France
F. Michel Dekking
Delft University of Technology
Faculty EEMCS
Delft
Netherlands