Open AccessBook
Fibonacci and Lucas Numbers with Applications
Reads0
Chats0
TLDR
The first 100 Lucas Numbers and their prime factorizations were given in this article, where they were shown to be a special case of the first 100 Fibonacci Numbers and Lucas Polynomials.Abstract:
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.read more
Citations
More filters
Journal ArticleDOI
Binomial Transforms of the Padovan and Perrin Matrix Sequences
Nazmiye Yilmaz,Necati Taskara +1 more
TL;DR: In this article, the binomial transforms are applied to Padovan and Perrin Matrices and the Binet formulas, summations, and generating functions of these transforms are found by recurrence relations.
Journal ArticleDOI
On Jacobsthal and Jacobsthal–Lucas Octonions
Cennet Bolat Çimen,Ahmet İpek +1 more
TL;DR: In this article, the Jacobsthal octonions and Jacob Sthal-Lucas octonion sequences were established and the relation between Jacobsthaels and Lucas octonsions was derived.
Journal ArticleDOI
Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence
István Mező,Ayhan Dil +1 more
TL;DR: In this paper, the Euler-Seidel method was used for deriving new identities for hyperharmonic and r-Stirling numbers, which is a generalization of Gosper's identity.
Journal ArticleDOI
A New Solution Concept for the Ultimatum Game leading to the Golden Ratio.
TL;DR: The optimality principle is explained both in an axiomatic way and by bargaining arguments, and the relation to Fibonacci numbers is outlined.
References
More filters
Journal ArticleDOI
Elementary number theory
TL;DR: The book as mentioned in this paper is designed for a first course in number theory with minimal prerequisites and is designed to stimulate curiosity about numbers and their properties, including almost a thousand imaginative exercises and problems.
Book
An introduction to the history of mathematics
TL;DR: In this paper, the authors present a Chronological Table of Cultural Connections: The Hunters of the Savanna (The Stone Age), Numeral Systems, the Agricultural Revolution (The Cradles of Civilization). Babylonian and Egyptian Mathematics.
Journal ArticleDOI
Unsolved problems in number theory
TL;DR: The topics covered are: additive representation functions, the Erdős-Fuchs theorem, multiplicative problems (involving general sequences), additive and multiplicative Sidon sets, hybrid problems (i.e., problems involving both special and general sequences, arithmetic functions and the greatest prime factor func- tion and mixed problems.
Book
Linear Algebra With Applications
TL;DR: Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra.