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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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Polynomials Generated by the Fibonacci Sequence
TL;DR: In this article, the polynomial Fibonacci sequence p is defined by setting p 0(x) = 1 and p 1 = 1, where x is the number of nodes in the p.
Journal ArticleDOI
On the k -Lucas Numbers and Lucas Polynomials
TL;DR: In this article, the authors introduce an operator in order to derive new symmetric properties of Lucas numbers and Lucas polynomials, and make use of this operator to give some new generating functions for Lucas numbers.
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A novel audio watermarking scheme based on fuzzy inference system in DCT domain
TL;DR: A new design of audio watermarking is introduced which is able to make compromise between transparency, robustness, and capacity by means of the synergy between fuzzy inference system, Singular Value Decomposition (SVD), and Fibonacci’s sequence in Discrete Cosine Transform (DCT).
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On Integer Numbers with Locally Smallest Order of Appearance in the Fibonacci Sequence
TL;DR: This paper will construct infinitely many natural numbers satisfying the previous inequalities and which do not belong to the Fibonacci sequence.
Journal ArticleDOI
On the spectral norms of r-circulant matrices with the biperiodic Fibonacci and Lucas numbers.
Cahit Köme,Yasin Yazlik +1 more
TL;DR: Lower and upper bounds are obtained for the spectral norms of Kronecker and Hadamard products of Q and L matrices whose entries are the bi Periodic Fibonacci and biperiodic Lucas numbers.
References
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Elementary number theory
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