Fibonacci and Lucas numbers with applications

Fibonacci and Lucas numbers have intrigued amateur and professional mathematicians for centuries. This volume represents the first attempt to compile a definitive history and authoritative analysis of these famous integer sequences, complete with a wealth of exciting applications, enlightening examples, and fun exercises that offer numerous opportunities for exploration and experimentation.

The author has assembled a myriad of fascinating properties of both Fibonacci and Lucas numbers–as developed by a wide range of sources–and catalogued their applications in a multitude of widely varied disciplines such as art, stock market investing, engineering, and neurophysiology. Most of the engaging and delightful material here is easily accessible to college and even high school students, though advanced material is included to challenge more sophisticated Fibonacci enthusiasts. A historical survey of the development of Fibonacci and Lucas numbers, biographical sketches of intriguing personalities involved in developing the subject, and illustrative examples round out this thorough and amusing survey. Most chapters conclude with numeric and theoretical exercises that do not rely on long and tedious proofs of theorems. Highlights include:

• Balanced blend of theory and real-world applications
• Excellent reference material for student reports and projects
• User-friendly, informal, and entertaining writing style
• Historical interjections and short biographies that add a richer perspective to the topic
• Reference sections providing important symbols, problem solutions, and fundamental properties from the theory of numbers and matrices

Table of Contents

• 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 Prime Factorizations.
• Appendix 4: The First 100 Lucas Numbers and Their Prime Factorizations.
• References.
• Solutions to Odd-Numbered Exercises.
• Index.