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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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Book ChapterDOI
Some weighted sums of products of Lucas sequences
Emrah Kılıç,Neşe Ömür +1 more
TL;DR: The weighted sums of products of Lucas sequences of the form n ∑ k=0 ( n k ) rmksm(tn+k) are considered and it is shown that these sums could be expressed via terms of the Lucas sequences.
Journal ArticleDOI
Regular matrix of interval numbers based on Fibonacci numbers
TL;DR: In this article, the authors introduced the criterion for the regularity of a matrix whose elements are interval numbers and defined a regular matrix of interval numbers using fibonacci numbers and introduced some new sequence spaces.
Quasicrystal Approximants in the RE-Au-SM systems (RE = Gd, Tb, Ho, Yb; SM = Si, Ge) : Syntheses, structures and properties
TL;DR: In this paper, Tsai-type 1/1 quasicrystal approximants were synthesized using high temperature synthesis techniques such as self-healing and self-decomposition.
Journal ArticleDOI
On the number of perfect matchings for some certain types of bipartite graphs
TL;DR: In this article, the authors consider the relationship between the number of perfect matchings (1-factors) for some certain types of bipartite graphs and Fibonacci and Lucas numbers.
Posted Content
Fibonacci Numbers, Statistical Convergence and Applications
Murat Kirişci,Ali Karaisa +1 more
TL;DR: In this article, the definition of new statistical convergence with Fibonacci sequence is given and some fundamental properties of statistical convergence are examined, and approximation theory worked as a application of the statistical convergence.
References
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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.