Parity and Primality of Catalan Numbers

In In the last decades, Steganography techniques have been applied in a variety of data files. The need of copyrightprotection in Music, in Photography e.t.c pushed the software companies to develop many steganographic systemswhich they use, in various areas, e.g., in digital assets (DRM). In this paper, we propose a number of methods forimage steganography using Catalan numbers and Lucas numbers and we show that they produce better resultsthan the technique using Fibonacci numbers. We are able to use Catalan and Lucas numbers since we haveproved that these sets of numbers satisfy similar conditions to those of the Theorem of Zeckendorf.

Lucas and Gibonacci numbers are two sequences of numbers derived from a welknown numbers, Fibonacci numbers. The difference between Lucas and Fibonacci numbers only lies on the first and second elements. The first element in Lucas numbers is 2 and the second is 1, and nth element, n ≥ 3 determined by similar pattern as in the Fibonacci numbers, i.e : Ln = Ln-1 + Ln-2. Gibonacci numbers G0 , G1 ,G2 , ...; Gn = Gn-1 + Gn-2 are generalized of Fibonacci numbers, and those numbers are nonnegative integers. If G0 = 1 and G1 = 1, then the numbers are the wellknown Fibonacci numbers, and if G0 = 2 and G1 = 1, the numbers are Lucas numbers. Thus, the difference of those three sequences of numbers only lies on the first and second of the elements in the sequences. For Fibonacci numbers there are quite a lot identities already explored, including the sum of cubes, but there have no discussions yet about the sum of cubes for Lucas and Gibonacci numbers. In this study the sum of cubes of Lucas and Gibonacci numbers will be discussed and showed that the sum of cubes for Lucas numbers is and for Gibonacci numbers is

Fibonacci and Lucas sequences are “two shining stars in the vast array of integer sequences,” and because of their ubiquitousness, tendency to appear in quite unexpected and unrelated places, abundant applications, and intriguing properties, they have fascinated amateurs and mathematicians alike. However, Catalan numbers are even more fascinating. Like the North Star in the evening sky, they are a beautiful and bright light in the mathematical heavens. They continue to provide a fertile ground for number theorists, especially, Catalan enthusiasts and computer scientists. Since the publication of Euler's triangulation problem (1751) and Catalan's parenthesization problem (1838), over 400 articles and problems on Catalan numbers have appeared in various periodicals. As Martin Gardner noted, even though many amateurs and mathematicians may know the abc's of Catalan sequence, they may not be familiar with their myriad unexpected occurrences, delightful applications, properties, or the beautiful and surprising relationships among numerous examples. Like Fibonacci and Lucas numbers, Catalan numbers are also an excellent source of fun and excitement. They can be used to generate interesting dividends for students, such as intellectual curiosity, experimentation, pattern recognition, conjecturing, and problem-solving techniques. The central character in the nth Catalan number is the central binomial coefficient. So, Catalan numbers can be extracted from Pascal's triangle. In fact, there are a number of ways they can be read from Pascal's triangle; every one of them is described and exemplified. This brings Catalan numbers a step closer to number-theory enthusiasts, especially.

The present COVID pandemic has transformed a physical world to a digital world. Electronic communication has become a major part of the human life that leads to a threat to digital network. So, hiding and protecting the information against unintended persons are highly essential nowadays. This can be done by encryption process. Encryption techniques are derived from mathematical concepts like number theory, graph theory, and algebra. The present paper explains a symmetric packet cipher using polygon triangulation and Catalan number of applied number theory. Here, a natural number n is secret between the users. The Catalan number Cn and number of triangles of n-angle Tn have major role in encryption process with simple logical XOR operation. To protect the cipher against different active and passive attacks, to achieve avalanche effect, the present plaintext packet is concatenated with the previous cipher text packet. KeywordsCatalan number CnPolygon triangulation TnEncryptionDecryption

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.

Just as Fibonacci and Lucas numbers are a great source of fun and excitement for both amateurs and professionals alike (Askey 2005, Koshy 2002), so are the less well-known Catalan numbers. They too are excellent candidates for mathematical activities such as experimentation, exploration, and conjecture. I was surprised to see that, like Fibonacci and Lucas numbers, Catalan numbers seemed to show up in several problems I had assigned to students over the years.

A Strahler bijection between Dyck paths and planar trees

Fibonaccene.- On a Class of Numbers Related to Both the Fibonacci and Pell Numbers.- A Property of Unit Digits of Fibonacci Numbers.- Some Properties of the Distributions of Order k.- Convolutions for Pell Polynomials.- Cyclotomy-Generated Polynomials of Fibonacci Type.- On Generalized Fibonacci Process.- Fibonacci Numbers of Graphs III: Planted Plane Trees.- A Distribution Property of Second-Order Linear Recurrences.- On Lucas Pseudoprimes which are Products of s Primes.- Fibonacci and Lucas Numbers and the Morgan-Voyce Polynomials in Ladder Networks and in Electric Line Theory.- Infinite Series Summation Involving Reciprocals of Pell Polynomials.- Fibonacci and Lucas Numbers and Aitken Acceleration.- On Sequences having Third-Order Recurrence Relations.- On the Solution of the Equation G n = P(x).- Distributions and Fibonacci Polynomials of Order k, Longest Runs, and Reliability of Consecutive-k-Out-Of-n : F Systems.- Fibonacci-Type Polynomials and Pascal Triangles of Order k.- A Note on Fibonacci and Related Numbers in the Theory of 2 x 2 Matrices.- Problems on Fibonacci Numbers and Their Generalizations.- Linear Recurrences having almost all Primes as Maximal Divisors.- On the Asymptotic Distribution of Linear Recurrence Sequences.- Golden Hops Around a Circle.

Fibonacci and Lucas numbers are special number sequences that have been the subject of many studies throughout history due to the relations they provide. The studies are continuing today, and findings about these number sequences are constantly increasing. The relations between the Fibonacci and Lucas numbers, which were found during the proof of the prime between two consecutive numbers belonging to the Fibonacci or Lucas number sequence with the Euclidean algorithm, started our project. In the project, Diophantine equations whose coefficients are Lucas or Fibonacci numbers have been studied, various relations have been found, and their proofs have been made. \begin{align*} F_nx-F_{n+1}y & =(-1)^n, \\ L_nx-L_{n-1}y & =1. \end{align*} As in the above example, the equivalents of $x$ and $y$ values were found in the Diophantine equations with Fibonacci and Lucas number coefficients; and based on this example, different variations of the Diophantine equations whose coefficients were selected from the Fibonacci and Lucas number sequences were created, and their proofs were made. Secondly, the geometric shapes consisting of vertices determined by pair of numbers selected from the Fibonacci or Lucas number sequence were considered, and their properties were examined. Various relations were found between them, and generalizations were made.

Binary unlabeled ordered trees (further called binary trees) were studied at least since Euler, who enumerated them. The number of such trees with n nodes is now known as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., height and path length distributions for a randomly selected tree). Binary trees find an abundance of applications in computer science. However, recently Seroussi posed a new and interesting problem motivated by information theory considerations: how many binary trees of a \emphgiven path length (sum of depths) are there? This question arose in the study of \emphuniversal types of sequences. Two sequences of length p have the same universal type if they generate the same set of phrases in the incremental parsing of the Lempel-Ziv'78 scheme since one proves that such sequences converge to the same empirical distribution. It turns out that the number of distinct types of sequences of length p corresponds to the number of binary (unlabeled and ordered) trees, T_p, of given path length p (and also the number of distinct Lempel-Ziv'78 parsings of length p sequences). We first show that the number of binary trees with given path length p is asymptotically equal to T_p ~ 2^2p/(log_2 p)(1+O(log ^-2/3 p)). Then we establish various limiting distributions for the number of nodes (number of phrases in the Lempel-Ziv'78 scheme) when a tree is selected randomly among all trees of given path length p. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the WKB method and matched asymptotics.

where n > 0 [2, 9]. Such numbers are now called Catalan numbers. However, Swiss mathematician Leonhard Euler (1707-1783) had previously encountered them around 1751 while investigating triangulations of convex polygons. In fact, Chinese mathematician Antu Ming (16927-1763?) discovered them even earlier, about 1730, through his geometric models. To our delight and at same time surprise, Catalan numbers occur in numerous seemingly unrelated places and situations, including combinatorics, abstract algebra, linear algebra, algebraic geometry, and sports [9]. They have same delightful propensity for popping up unexpectedly, particularly in combinatorial problems, Mar tin Gardner wrote in 1976 in his popular column Mathematical games in Scientific American [5]. Indeed, he adds, the Catalan sequence is probably most fre quently encountered sequence that is still obscure enough to cause mathematicians lacking access to N. J. A. Sloane's A Handbook of Integer Sequences to expend inor dinate amounts of energy re-discovering formulas that were worked out long ago. For example, Stanley lists over 70 occurrences of Catalan numbers in his book [11] and another 70 on his website Catalan Addendum.

A symmetric algorithm for hyperharmonic and Fibonacci numbers

Fibonacci numbers, Lucas numbers and Mulatu numbers are built in the same method. The three numbers differ in the first term, while the second term is entirely the same. The next terms are the sum of two successive terms. In this article, generalizations of Fibonacci, Lucas and Mulatu (GFLM) numbers are built which are generalizations of the three types of numbers. The Binet formula is then built for the GFLM numbers, and determines the golden ratio, silver ratio and Bronze ratio of the GFLM numbers. This article also presents generalizations of these three types of ratios, called Metallic ratios. In the last part we state the Metallic ratio in the form of continued fraction and nested radicals.

Objective: Classical orthogonal polynomials are extensively used for the numerical solution of differential equations and numerical analysis. Various generalization problems involving Chebyshev polynomials have been studied and one such area is the classical linearization problem. In this paper, we will consider a generalized classical linearization problem and study some identities representing the sums of finite products of the 3rd and 4th kinds of Chebyshev polynomials, Fibonacci numbers, and Lucas numbers as a linear sum of the Chebyshev polynomials of 2nd kind and their derivatives. Methods: Differential calculus, combinatorial, and elementary algebraic computations are employed to yield the results. Findings: A new set of identities expressing the sums of finite products of the 3rd and 4th kinds of Chebyshev polynomials, Fibonacci numbers, and Lucas numbers as a linear sum of the 2nd kinds of Chebyshev polynomials and their derivatives are studied. Novelty: Earlier studies have established the sums of finite products of the Lucas numbers, Fibonacci numbers, Fibonacci and Pell polynomials, and Chebyshev polynomials of the 1st and 2nd kinds as a linear sum of other orthogonal polynomials. However, representations of sums of finite products of the 3rd and 4th kinds of Chebyshev polynomials, Fibonacci numbers, and Lucas numbers as a linear sum of the 2nd kinds of Chebyshev polynomials and their derivatives have not been studied, which is the prime focus of this manuscript. Keywords: Linearization, Chebyshev Polynomials, Fibonacci Numbers, Lucas numbers, Orthogonality

In this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equationsxn+1=1+2yn−k3+yn−k,yn+1=1+2zn−k3+zn−k,zn+1=1+2xn−k3+xn−k,$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$wheren,k∈ ℕ0, the initial valuesx−k,x−k+1, …,x0,y−k,y−k+1, …,y0,z−k,z−k+1, …,z1andz0are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.