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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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The fibonacci numbers-exposed
Dan Kalman,Robert Mena +1 more
TL;DR: The Lucas numbers as discussed by the authors are a close relative of the Fibonacci numbers and have achieved a kind of celebrity status, referred to as the "two shining stars in the vast array of integer sequences".
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On Fibonacci Quaternions
TL;DR: In this paper, the authors investigated the Fibonacci and Lucas quaternions and derived the generating functions and Binet formulas for these two types of quadratic functions. And they also derived some sums formulas for them.
References
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Book ChapterDOI
Inverse Trigonometric Functions
TL;DR: In this paper, the authors provide an overview on inverse trigonometric functions, and illustrate the corresponding graphs of the trigonometrical functions, including arcsin, arctan, arccos, and arccot.
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Fibonacci Numbers and Geometry
TL;DR: In this article, a unit segment AB (see Figure 2) is broken into two pieces in such a way that the greater part is the mean proportional between the smaller part and the whole segment.
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On the non-existence of Fibonacci squares
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