Table Of ContentProblem Books in Mathematics
Dorin Andrica
Ovidiu Bagdasar
Recurrent
Sequences
Key Results, Applications, and Problems
Problem Books in Mathematics
SeriesEditor
PeterWinkler
DepartmentofMathematics
DartmouthCollege
Hanover,NH
USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/714
Dorin Andrica • Ovidiu Bagdasar
Recurrent Sequences
Key Results, Applications, and Problems
123
DorinAndrica OvidiuBagdasar
DepartmentofMathematics CollegeofEngineeringandTechnology
“Babes¸-Bolyai”University UniversityofDerby
Cluj-Napoca,Romania Derby,UK
ISSN0941-3502 ISSN2197-8506 (electronic)
ProblemBooksinMathematics
ISBN978-3-030-51501-0 ISBN978-3-030-51502-7 (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
©SpringerNatureSwitzerlandAG2020
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
“YourdaysincreasewitheachTomorrow” (“Cumâinezilele-t¸iadaogi”)
byMihaiEminescu
YourdaysincreasewitheachTomorrow,
YourlifegrowslesswithYesterday,
Infrontofyoutherelies,however,
Foralloftheeternity:Today.
These famous verses of the national poet of the Romanians capture the close
connections between poetry and mathematics. Indeed, if D denotes the number
n
of days in someone’s life, then the equation of life described by the poet can be
encodedinthefollowingwell-knownrecurrence:
Dn+1−Dn−1 =Dn,
whichcoincideswiththerecurrencesatisfiedbytheFibonaccinumbers.
Preface
Overviewand Goals
This book presents the state-of-the-art results concerning recurrent sequences and
their practical applications in algebra, number theory, geometry of the complex
plane, discrete mathematics, or combinatorics. The purpose of this book is to
familiarize more readers with recent developments in this area and to encourage
furtherresearch.
The content of the book is recent and reflects current research in the field
of recurrent sequences. Some new approaches promoted by this book are visual
representationsofrecurrencesinthecomplexplaneandtheuseofmultiplemethods
forderivingnovelidentitiesornumbersequences.
Thefirstpartofthebookisdedicatedtofundamentalresultsandkeyexamplesof
recurrencesandtheirproperties.Wealsopresentthegeometryoflinearrecurrences
in the complex plane in some detail. Here, some recent research has led to
unexpecteddevelopmentswithincombinatorics,numbertheory,integersequences,
andrandomnumbergeneration.
Thesecond partofthebook presents123olympiad trainingproblemswithfull
solutions and some appendices. These relate to linear recurrences of first, second,
andhigherorders,classicalsequences,homographicrecurrences,systemsofrecur-
rences,complexrecurrentsequences,andrecurrentsequencesincombinatorics.
Audience
This book will be useful for researchers and scholars who are interested in recent
advances in the field of recurrences, postgraduate students in college or university
andtheirinstructors,oradvancedhighschoolstudents.
Students training for mathematics competitions and their coaches will find
numerous worked examples and problems with detailed solutions. Many of these
vii
viii Preface
problems are original, while others are selected from international olympiads or
fromvariousspecialistjournals.
Organizationand Features
The book is organized in eight chapters and an appendix. The first six chapters
are dedicated to theoretical results, examples, and applications, while the last two
containolympiadproblemsaccompaniedbyfullsolutions.
Chapter1presentsthefundamentalaspectsconcerningrecurrencerelations,such
as explicit and implicit forms, order, systems of recurrent sequences, or existence
anduniquenessofthesolution.Thechapteralsopresentssomerecurrencesequences
arising in mathematical modeling, algebra, combinatorics, geometry, analysis, or
iterativenumericalmethods.
Chapter2isdedicatedtofirst-andsecond-orderlinearrecurrenceswithgeneral
coefficients, various classical sequences and polynomials (including Fibonacci,
Lucas, Pell, or Lucas–Pell), and homographic recurrences defined by linear frac-
tionaltransformationsinthecomplexplane.
Chapter 3 presents the arithmetic properties and trigonometric formulae for
classical recurrent sequences, extending some results for Fibonacci and Lucas
numbers.
Chapter 4 is dedicated to ordinary and exponential generating functions for
classical functions and polynomials and presents numerous new results. It also
presentssomenewusefulversionsofCauchy’sintegralformula.
Chapter 5 explores the dynamics of second-order linear recurrences in the
complex plane, referred to as Horadam sequences. We first formulate periodicity
conditions, which are then used to investigate the geometric structure and the
numberofperiodicHoradampatterns.Anatlasofnon-periodicHoradampatternsis
alsopresented,followedbyanapplicationtopseudo-randomnumbergenerators.We
concludewithsomeexamplesofperiodicnonhomogeneousHoradamsequences.
Chapter6presentsthedynamicsofcomplexlinearrecurrentsequencesofhigher
order and investigates the periodicity, geometric structure, and enumeration of the
periodic patterns. Some results concerning systems of linear recurrence sequences
arealsoanalyzed,togetherwithapplicationstoDiophantineequations.Anatlasof
complex linear recurrent patterns (both periodic and non-periodic) for third-order
recurrencesisalsodiscussed,alongwithconnectionstofinitedifferences.
Chapter 7 contains 123 olympiad training problems involving recurrent
sequences, solved in detail in Chapter 8. Many problems are original and concern
linear recurrence sequences of first, second, and higher orders, some classical
sequences, homographic sequences, systems of sequences, complex recurrence
sequences,andrecursionsincombinatorics.
Preface ix
Prerequisites
Our intention was to make the book as self-contained as possible, so we have
included many definitions together with examples. Chapters 1, 2, and 5 only
require good understanding of college algebra, complex numbers, analysis, and
basic combinatorics. For Chapters 3, 4, and 6, the prerequisites include number
theory,linearalgebra,andcomplexanalysis.
HowtoUsetheBook
The book presents recurrent sequences in connection with a wide range of topics
andisillustratedwithnumerousexamplesanddiagrams.
The introductory chapter and some of the properties of first- and second-order
linear recurrences (which include classical number sequences) are elementary and
canbeunderstoodbyhighschoolstudents.
More advanced topics such as homographic recurrences, generating functions,
higher-order recurrent sequences, or systems of recurrent sequences require con-
ceptsusuallycoveredincollegeorundergraduatecourses.
We have also included many original and recent results on arithmetic and
trigonometric properties of classical sequences, new applications of Cauchy’s
integral formula in the study of polynomial coefficients, or the complex recurrent
patternsappliedinpseudo-randomnumbergeneration,whichexposethereaderto
thestateoftheartinthefield.
The collection of problems with full solutions illustrates the wide range of
topicswhererecurrentsequencescanbefoundandrepresentsanidealmaterialfor
preparingstudentsforOlympiads.
Thebookalsoincludes177referencesandanindexwhichwillhelpthereaders
tofurtherinvestigatekeynotionsandconcepts.
Acknowledgements
We would like to thank Dr George Ca˘ta˘lin T¸urcas¸ and Remus Miha˘ilescu, for
checkingthedocumentcarefullyandprovidingfeedback.
We are indebted to the anonymous referees for their constructive remarks and
suggestions,whichhelpedustoimprovethequalityofthemanuscript.
Specialthankstoourfamilies,fortheircontinuousanddiscretesupport.
Cluj-Napoca,Romania DorinAndrica
Derby,UK OvidiuBagdasar
May2020
Contents
1 IntroductiontoRecurrenceRelations ..................................... 1
1.1 RecursiveSequencesofOrderk....................................... 2
1.2 RecurrentSequencesDefinedbyaSequenceofFunctions .......... 3
1.3 SystemsofRecurrentSequences ...................................... 3
1.4 ExistenceandUniquenessoftheSolution ............................ 4
1.5 RecurrentSequencesArisinginPracticalProblems.................. 6
1.5.1 ApplicationsinMathematicalModeling ..................... 6
1.5.2 Algebra......................................................... 8
1.5.3 Combinatorics ................................................. 9
1.5.4 Geometry....................................................... 11
1.5.5 Analysis........................................................ 12
1.5.6 IterativeNumericalMethods.................................. 14
2 BasicRecurrentSequences ................................................. 19
2.1 First-OrderLinearRecurrentSequences.............................. 19
2.2 Second-OrderLinearRecurrentSequences........................... 26
2.2.1 HomogeneousRecurrentSequences.......................... 26
2.2.2 NonhomogeneousRecurrentSequences ..................... 34
2.2.3 Fibonacci,Lucas,Pell,andPell–LucasNumbers............ 38
2.2.4 ThePolynomialsU (x,y)andV (x,y) ..................... 48
n n
2.2.5 PropertiesofFibonacci,Lucas,Pell,andLucas–Pell
Numbers ....................................................... 58
2.2.6 Zeckendorf’sTheorem........................................ 65
2.3 HomographicRecurrences............................................. 67
2.3.1 KeyDefinitions................................................ 67
2.3.2 HomographicRecurrentSequences .......................... 68
2.3.3 RepresentationTheoremsforHomographicSequences..... 72
2.3.4 ConvergenceandPeriodicity.................................. 74
2.3.5 HomographicRecurrenceswithVariableCoefficients ...... 77
xi
