Koshy Fibonacci and Lucas Numbers with Applications

1. Auflage 2011
ISBN: 978-1-118-03131-5
Verlag: John Wiley & Sons
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 672 Seiten, E-Book

Reihe: Wiley Series in Pure and Applied Mathematics

The first comprehensive survey of mathematics' most fascinatingnumber sequences
Fibonacci and Lucas numbers have intrigued amateur and professionalmathematicians for centuries. This volume represents the firstattempt to compile a definitive history and authoritative analysisof these famous integer sequences, complete with a wealth ofexciting applications, enlightening examples, and fun exercisesthat offer numerous opportunities for exploration andexperimentation.
The author has assembled a myriad of fascinating properties of bothFibonacci and Lucas numbers-as developed by a wide range ofsources-and catalogued their applications in a multitude of widelyvaried disciplines such as art, stock market investing,engineering, and neurophysiology. Most of the engaging anddelightful material here is easily accessible to college and evenhigh school students, though advanced material is included tochallenge more sophisticated Fibonacci enthusiasts. A historicalsurvey of the development of Fibonacci and Lucas numbers,biographical sketches of intriguing personalities involved indeveloping the subject, and illustrative examples round out thisthorough and amusing survey. Most chapters conclude with numericand theoretical exercises that do not rely on long and tediousproofs of theorems. Highlights include:
* Balanced blend of theory and real-world applications
* Excellent reference material for student reports andprojects
* User-friendly, informal, and entertaining writing style
* Historical interjections and short biographies that add a richerperspective to the topic
* Reference sections providing important symbols, problemsolutions, and fundamental properties from the theory of numbersand matrices
Fibonacci and Lucas Numbers with Applications providesmathematicians with a wealth of reference material in oneconvenient volume and presents an in-depth and entertainingresource for enthusiasts at every level and from any background.

Autoren/Hrsg.

Koshy, Thomas

Inhaltsverzeichnis

Preface.
List of Symbols.
Leonardo Fibonacci.
The Rabbit Problem.
Fibonacci Numbers in Nature.
Fibonacci Numbers: Additional Occurrances.
Fibonacci and Lucas Identities.
Geometric Paradoxes.
Generalized Fibonacci Numbers.
Additional Fibonacci and Lucas Formulas.
The Euclidean Algorithm.
Solving Recurrence Relations.
Completeness Theorems.
Pascal's Triangle.
Pascal-Like Triangles.
Additional Pascal-Like Triangles.
Hosoya's Triangle.
Divisibility Properties.
Generalized Fibonacci Numbers Revisited.
Generating Functions.
Generating Functions Revisited.
The Golden Ratio.
The Golden Ratio Revisited.
Golden Triangles.
Golden Rectangles.
Fibonacci Geometry.
Regular Pentagons.
The Golden Ellipse and Hyperbola.
Continued Fractions.
Weighted Fibonacci and Lucas Sums.
Weighted Fibonacci and Lucas Sums Revisited.
The Knapsack Problem.
Fibonacci Magic Squares.
Fibonacci Matrices.
Fibonacci Determinants.
Fibonacci and Lucas Congruences.
Fibonacci and Lucas Periodicity.
Fibonacci and Lucas Series.
Fibonacci Polynomials.
Lucas Polynomials.
Jacobsthal Polynomials.
Zeros of Fibonacci and Lucas Polynomials.
Morgan-Voyce Polynomials.
Fibonometry.
Fibonacci and Lucas Subscripts.
Gaussian Fibonacci and Lucas Numbers.
Analytic Extensions.
Tribonacci Numbers.
Tribonacci Polynomials. Appendix 1: Fundamentals.
Appendix 2: The First 100 Fibonacci and Lucas Numbers.
Appendix 3: The First 100 Fibonacci Numbers and Their PrimeFactorizations.
Appendix 4: The First 100 Lucas Numbers and Their PrimeFactorizations.
References.
Solutions to Odd-Numbered Exercises.
Index.

Über Autor(innen)

THOMAS KOSHY, Ph.D., is Professor of Mathematics at Framingham State College in Framingham, Massachusetts. He is the author of the forthcoming book Elementary Number Theory with Applications, among several other books, as well as numerous articles across a wide range of topics.