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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
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New presentations for real numbers
Nihal Yılmaz Özgür,Sümeyra Uçar +1 more
TL;DR: In this paper, it was shown that every real number can be uniquely represented as the sum of the squares of consecutive kFibonacci numbers, and presented new presentation theorems for any real number u 6 = 0 using kFIBonacci series.
On a sequence of tridiagonal matrices, whose permanents are related to fibonacci
TL;DR: In this article, it was shown that more general sequences of tridiagonal matrices with Fibonacci numbers are related to the sequence of connection permanents of the same matrices.
Journal ArticleDOI
Circular-hyperbolic Fibonacci quaternions
TL;DR: In this paper, some algebraic properties of circular-hyperbolic Fibonacci numbers and quaternions which are connected with circular hyperbolic numbers and Fibonaci numbers are investigated, including Honsberger's identity, the generating function, Binet's formula, d'Ocagne's identity and Cassini's identity.
Posted Content
(p,q)−deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms
TL;DR: A characterization of (p,q)-deformed Fibonacci and Lucas polynomi- als is given in this article, where their generating functions and Fourier integral transforms are explicitly computed and discussed.
Journal ArticleDOI
Determinants, inverses, norms, and spreads of skew circulant matrices involving the product of Fibonacci and Lucas numbers
TL;DR: In this paper, the invertibility of n× n skew circulant matrix involving the product of Fibonacci and Lucas numbers was investigated, whose determinant and inverse can be expressed by the (n− 1)th, nth, (n+ 1)st, n+ 2)th product of the two numbers.
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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An introduction to the history of mathematics
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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.