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
On the Universal Regularity of the Numbers of Generalized Recurrence Sequence and Solutions to Its Characteristic Equation of Second Order
TL;DR: In this article, it was shown that the classical oscillations of the ratio of neighboring members of the Fibonacci sequences are valid for arbitrary directions on the plane of the phase coordinates, approaching, to a maximum, the solutions to the characteristic quadratic equation at a given point.
Journal ArticleDOI
On solutions of the diophantine equation $$\varvec{F_{n}-F_{m}=3^{a}}$$ F n - F m = 3 a
TL;DR: In this article, the authors proved non-negative integer solutions of the diophantine equation for n, m, a and a for 3-a, where n,m,a are Fibonacci numbers.
Posted Content
Optimal cube factors of Fibonacci and matchable Lucas cubes
Xu Wang,Xuxu Zhao,Haiyuan Yao +2 more
TL;DR: In this paper, the optimal cube factor of a graph, a special kind of component factor, is introduced and some results on the Padovan sequence and binomial coefficients are obtained.
Posted Content
An Identity for Second Order Sequences Obeying the Same Recurrence Relation
Abstract: We derive an identity connecting any two second-order linear recurrence sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are developed. Illustrative examples are drawn from Fibonacci, Fibonacci-Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas sequences and their generalizations. Our new results subsume previously known identities.
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.