Fibonacci and Lucas Numbers with Applications (Volume 1 and 2)

Fibonacci and Lucas Numbers with Applications provides a user-friendly and historical approach to the many fascinating properties of Fibonacci and Lucas numbers, which have intrigued amateurs and professionals for centuries. Offering an in-depth study of the topic, this book includes exciting applications that provide many opportunities to explore and experiment.

Introduction:

The Fibonacci sequence and the Lucas sequence are two very bright shining stars in the vast array of integer sequences. They have fascinated amateurs, and professional architects, artists, biologists, musicians, painters, photographers, and mathematicians for centuries; and they continue to charm and enlighten us with their beauty, their abundant applications, and their ubiquitous habit of occurring in totally surprising and unrelated places. They continue to be a fertile ground for creative amateurs and mathematicians alike, and speak volumes about the vitality of this growing field.

This book originally grew out of my fascination with the intriguing beauty and rich applications of the twin sequences. It has been my long-cherished dream to study and to assemble the myriad properties of both sequences, developed over the centuries, and to catalog their applications to various disciplines in an orderly and enjoyable fashion. As the cryptanalyst Sophie Neveu in Dan Brown’s bestseller The Da Vinci Code claims, “the [Fibonacci] sequence … happens to be one of the most famous mathematical progressions in history.”

An enormous wealth of information is available in the mathematical literature on Fibonacci and Lucas numbers; but, unfortunately, most of it continues to be widely scattered in numerous journals, so it is not easily accessible to many, especially to non-professionals. The first edition was the end-product of materials collected and presented from a wide range of sources over the years; and to the best of my knowledge, it was the largest comprehensive study of this beautiful area of human endeavor.

So why this new edition? Since the publication of the original volume, I have had the advantage and fortune of hearing from a number of Fibonacci enthusiasts from around the globe, including students. Their enthusiasm, support, and encouragement were really overwhelming. Some opened my eyes to new sources and some to new charming properties; and some even pointed out some inexcusable typos, which eluded my own eyes. The second edition is the byproduct of their ardent enthusiasm, coupled with my own.

Many Fibonacci enthusiasts and amateurs know the basics of Fibonacci and Lucas numbers. But there are a multitude of beautiful properties and applications that may be less familiar. Fibonacci and Lucas numbers are a source of great fun; teachers and professors often use them to generate excitement among students, who find them stimulating their intellectual curiosity and sharpening their mathematical skills, such as pattern recognition, conjecturing, proof techniques, and problem-solving. In the process, they invariably appreciate and enjoy the beauty, power, and ubiquity of the Fibonacci family.

Contents:

Volume 1:

• Fibonacci Numbers
• Fibonacci Numbers in Nature
• Additional Fibonacci and Lucas Occurrences
• Fibonacci and Lucas Identities
• Geometric Illustrations and Paradoxes
• Gibonacci Numbers
• Additional Fibonacci and Lucas Formulas
• The Euclidean Algorithm
• Divisibility Properties
• Pascal’s Triangle
• Pascal-like Triangles
• Recurrences and Generating Functions
• Combinatorial Models I
• Hosoya’s Triangle
• The Golden Ratio
• Golden Triangles and Rectangles
• Figeometry
• Continued Fractions
• Fibonacci Matrices
• Graph-theoretic Models I
• Fibonacci Determinants
• Fibonacci and Lucas Congruences
• Fibonacci and Lucas Series
• Weighted Fibonacci and Lucas Sums
• Fibonometry I
• Completeness Theorems
• The Knapsack Problem
• Fibonacci and Lucas Subscripts
• Fibonacci and the Complex Plane

Volume 2:

• Fibonacci and Lucas Polynomials I
• Fibonacci and Lucas Polynomials II
• Combinatorial Models II
• Graph-Theoretic Models II
• Gibonacci Polynomials
• Gibonacci Sums
• Additional Gibonacci Delights
• Fibonacci and Lucas Polynomials III
• Gibonacci Determinants
• Fibonometry II
• Chebyshev Polynomials
• Chebyshev Tilings
• Bivariate Gibonacci Family I
• Jacobsthal Family
• Jacobsthal Tilings and Graphs
• Bivariate Tiling Models
• Vieta Polynomials
• Bivariate Gibonacci Family II
• Tribonacci Polynomials

Fibonacci and Lucas Numbers with Applications (Volume 1 and 2) By Thomas Koshy pdf