Thomas Koshy

• Framingham State University

We investigate the combinatorially defined Lobb numbers with some new relationships to the ubiquitous Catalan numbers.

We introduce the family of Gibonacci polynomials. Using the concept of the weight of a Fibonacci walk, we present a combinatorial interpretation of Fibonacci polynomials and derive a few well-known Fibonacci polynomial identities. Finally, we establish a bijection between the set of Fibonacci walks of length n and the set of Fibonacci tilings of a...

We extend the well-known Gelin-Cesàro identity Fn+2Fn+1Fn-1Fn-2 - Fn4 =1 to the Gibonacci family. This generalization has interesting consequences.

We present two sets of graph-theoretic models for Pell and Pell-Lucas numbers by constructing rooted binary trees.

We extend the well-known Lucas identity F3n+1 + F3n - F3n-1 = F3n and the Ginsburg identity F3n+2 - 3F3n + F3n-2 = 3F3n to Fibonacci and Lucas polynomials. This yields interesting dividends to Pell and Pell-Lucas polynomials and numbers.

Pell and Pell-Lucas numbers, like the well-known Fibonacci and Catalan numbers, continue to intrigue the mathematical world with their beauty and applicability. They offer opportunities for experimentation, exploration, conjecture, and problem-solving techniques, connecting the fields of analysis, geometry, trigonometry, and various areas of discre...

We employ a Vandermonde-like technique to develop a combinatorial identity and then use it to establish a new result. We then deduce three Catalan formulas from the latter identity. Catalan numbers; Vandermonde's identity.

We investigate some divisibility properties of Catalan numbers with Mersenne numbers Mk as their subscripts, and then extend them for k = Fn, Ln, Pn, Qn, and Cn, where Fn, Ln, Pn, Qn, and C n denote the nth Fibonacci, Lucas, Pell, Pell-Lucas, and Catalan numbers, respectively. We also compute CMk moduli 4, 8, and 64; and CMk+1 modulo 128.

Using generating functions, we develop a number of properties of sums of products of Fibonacci, Lucas, Pell, Pell–Lucas, Jacobsthal and Jacobsthal–Lucas numbers.

96.46 Some Catalan identities with interesting consequence - Volume 96 Issue 536 - Thomas Koshy

The well known Catalan numbers Cn are named after Belgian mathematician Eugene Charles Catalan (1814–1894), who found them in his investigation of well-formed sequences of left and right parentheses. As Martin Gardner (1914–2010) wrote in Scientific American [2], they have the propensity to “pop up in numerous and quite unexpected places.” They occ...

In 1838, the Belgian mathematician Eugene C. Catalan (1814-1894) discovered that the number C n of well-fonned sequences, with n pairs of left and right parentheses, is given by where n > 0 [1, 2]. For example, there are exactly five well-formed sequences with three pairs of left and right parentheses: ()()(), ()(()), (())(), (()()), ((())). The...

This article investigates the numbers , originally studied by Catalan. We re-confirm that they are indeed integers. Using the close relationship between them and the Catalan numbers C n , we develop some divisibility properties for C n . In particular, we establish that , where f k denotes the kth Fermat number and M k the kth Mersenne number 2 k  ...

Using congruences, second-order Diophantine equations, and linear algebra, we identify Jacobsthal and Jacobsthal-Lucas numbers that are also triangular numbers.

95.13 Some divisibility properties of Catalan numbers - Volume 95 Issue 532 - Thomas Koshy, Zhenguang Gao

We develop a recurrence satisfied by the Fibonacci and Pell families. We then use it to find explicit formulae and generating functions for the hybrids FnPn, LnPn, FnQn and LnQn, where Fn, Ln, Pn and Qn denote the nth Fibonacci, Lucas, Pell and Pell–Lucas numbers, respectively. The recurrence enables us to find such relations for and .

Using differential calculus, and the generating functions for the central binomial coefficients 2n n and odd-numbered Catalan numbers C 2n+1 , we develop formulas for weighted sums ∑ k=1 n kC 2n+1 C 2n-2k+1 and ∑ k=1 n k 2 C 2n+1 C 2n-2k+1

The concept of the ordinary binomial coefficient can be employed to construct an interesting family of positive integers. Such a family was introduced around 1974 by W. Hansell using the triangular numbers where we call them tribinomial coefficients since they are binomial coefficients for triangular numbers. To this end, first we define correspon...

A. Lobb discovered an interesting generalization of Catalan's parenthesization problem, namely: Find the number L(n, m) of arrangements of n + m positive ones and n − m negative ones such that every partial sum is nonnegative, where 0 ≤ m ≤ n. This article uses Lobb's formula, L(n, m) = (2m + 1)/(n + m + 1) C(2n, n + m), where C is the usual binomi...

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....

92.61 Pythagorean triples with Pell generators - Volume 92 Issue 525 - Thomas Koshy

90.59 An intriguing relationship between Fibonacci and Jacobsthal polynomials - Volume 90 Issue 519 - Mohammad Salmassi, Thomas Koshy

90.50 Lattice points in a family of hyperbolas - Volume 90 Issue 518 - Thomas Koshy

90.01 Cassini-like formulas for Jacobsthal and Koshy polynomials - Volume 90 Issue 517 - Thomas Koshy, Mohammad Salmassi

The study of finite-state machines began with the neural networks investigations of Warren S. McCulloch and Walter Pitts in 1943. Nowadays paradigms of finite-state constructs can be seen everywhere such as in turnstiles, traffic signal controllers, automated teller machines, automated telephone service, garage door openers, household appliances, a...

Trees are the most important class of graphs that make fine modeling tools. Trees are widely used in mathematics and computer science, as well as in linguistics and the social sciences. This chapter presents the concept of a tree and the two necessary and sufficient conditions for a graph to be a tree. A tree is a connected acyclic graph. The chapt...

This approachable text studies discrete objects and the relationsips that bind them. It helps students understand and apply the power of discrete math to digital computer systems and other modern applications. It provides excellent preparation for courses in linear algebra, number theory, and modern/abstract algebra and for computer science courses...

Additional Fibonacci-Based PuzzlesAn Intriguing Sequence