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Fibonacci and Lucas Numbers with Applications
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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
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An application of Fibonacci numbers into infinite Toeplitz matrices
Emrah Evren Kara,Metin Başarır +1 more
TL;DR: The main purpose of as mentioned in this paper is to define a new regular matrix by us- ing Fibonacci numbers and investigate its matrix domain in the classical sequence spaces 'p,1,c and c0, where 1 p < 1.
Journal ArticleDOI
Some new paranormed difference sequence spaces derived by fibonacci numbers
Serkan Demiriz,Emrah Evren Kara +1 more
TL;DR: In this article, a paranormed sequence space derived by the sequences of Fibonacci numbers is defined, and the matrix transformations from these spaces to the Maddox's spaces are characterized.
Journal ArticleDOI
Dual Fibonacci Quaternions
TL;DR: In this paper, the dual Lucas quaternion and dual Fibonacci quaternions were defined and the Binet and Cassini formulas for these quaternIONS were given.
Pokroky matematiky, fyziky a astronomie
TL;DR: Volf et al. as mentioned in this paper presented the results, achieved by representatives of the Czech Republic, of the 21st century European Union Science Olympiad (EUSO), which has not even reached ten years.
Journal ArticleDOI
On the properties of Lucas numbers with binomial coefficients
TL;DR: Some new properties of Lucas numbers with binomial coefficients have been obtained to write Lucas sequences in a new direct way and some important consequences related to the Fibonacci numbers have been given.
References
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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.
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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.
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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.