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2019 Generalized Beatty sequences and complementary triples
Jean-Paul Allouche, F. Michel Dekking
Mosc. J. Comb. Number Theory 8(4): 325-341 (2019). DOI: 10.2140/moscow.2019.8.325

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 Vi=(pinα+qin+ri), i=1,2,3 that are complementary in the same sense.

Citation

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Jean-Paul Allouche. F. Michel Dekking. "Generalized Beatty sequences and complementary triples." Mosc. J. Comb. Number Theory 8 (4) 325 - 341, 2019. https://doi.org/10.2140/moscow.2019.8.325

Information

Received: 28 December 2018; Revised: 22 May 2019; Accepted: 5 June 2019; Published: 2019
First available in Project Euclid: 29 October 2019

zbMATH: 07126246
MathSciNet: MR4026541
Digital Object Identifier: 10.2140/moscow.2019.8.325

Subjects:
Primary: 11B83, 11B85, 11D09, 11J70, 68R15

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.8 • No. 4 • 2019
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