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Dorin Andrica Ovidiu Bagdasar

Recurrent Sequences

Key Results, Applications, and Problems

Problem Books in Mathematics Series Editor Peter Winkler Department of Mathematics Dartmouth College Hanover, NH USA

More information about this series at http://www.springer.com/series/714

Dorin Andrica • Ovidiu Bagdasar

Recurrent Sequences Key Results, Applications, and Problems

123

Dorin Andrica Department of Mathematics “Babe¸s-Bolyai” University Cluj-Napoca, Romania

Ovidiu Bagdasar College of Engineering and Technology University of Derby Derby, UK

ISSN 0941-3502 ISSN 2197-8506 (electronic) Problem Books in Mathematics ISBN 978-3-030-51501-0 ISBN 978-3-030-51502-7 (eBook) https://doi.org/10.1007/978-3-030-51502-7 Mathematics Subject Classification: 05A18, 11B39, 11B83, 11D45, 11N37, 11N56, 11N64, 11Y55, 11Y70, 40A05, 40A10 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

“Your days increase with each Tomorrow”

(“Cu mâine zilele-¸ti adaogi”) by Mihai Eminescu

Your days increase with each Tomorrow, Your life grows less with Yesterday, In front of you there lies, however, For all of the eternity: Today. These famous verses of the national poet of the Romanians capture the close connections between poetry and mathematics. Indeed, if Dn denotes the number of days in someone’s life, then the equation of life described by the poet can be encoded in the following well-known recurrence: Dn+1 − Dn−1 = Dn , which coincides with the recurrence satisfied by the Fibonacci numbers.

Preface

Overview and Goals This book presents the state-of-the-art results concerning recurrent sequences and their practical applications in algebra, number theory, geometry of