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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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Journal ArticleDOI
Hyperbolic fibonacci sequence
TL;DR: In this article, the authors investigated the hyperbolic Fibonacci sequence and the Hyperbolic FIFO numbers and gave recurrence relations, the golden ratio, and Binet's formula for the hyper-bolic fIFO sequence and Lorentzian inner product, cross product and mixed product.
On Fibonacci and Lucas Numbers of the Form cx 2
Refik Keskin,Zafer Yosma +1 more
TL;DR: In this paper, by using some congruences concerning with Fibonacci and Lucas numbers, they completely solved the Diophantine equations Ln = 2Lmx 2, Fn = 2Fmx 2, Ln= 6Lmx2, Fn = 3Fmx2, and Fn = 6FMX 2.
Journal ArticleDOI
On the Negatively Subscripted Padovan and Perrin Matrix Sequences
Nazmiye Yilmaz,Necati Taskara +1 more
TL;DR: By taking into account matrix properties for these new matrix sequences, some behaviours of negatively subscripted Padovan and Perrin numbers have been investigated and the important relationships between negatively subscripting Padovanand Perrin matrix sequences are presented.
$k$-FIBONACCI AND $k$-LUCAS GENERALIZED QUATERNIONS
TL;DR: In this paper, the authors investigated the properties of $k-$Fibonacci and $k-Lucas quaternions over the generalized quaternion algebra and derived several well-known identities such as Catalan's, Cassini's and d'Ocagne's identities.
Journal ArticleDOI
Cryptography using generalized Fibonacci matrices with Affine-Hill cipher
Kalika Prasad,Hrishikesh Mahato +1 more
TL;DR: A public key cryptography using Affine-Hill cipher with a generalized Fibonacci matrix called multinacci matrix is proposed, which reduces the time complexity as well as space complexity and comes with a large key-space.
References
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Unsolved problems in number theory
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Linear Algebra With Applications
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