Related papers: Honeycombs in the Pascal triangle and beyond
This paper presents new identities expressing the terms of Fibonacci, Lucas, and generalized Fibonacci sequences with multiple indices through powers of Lucas numbers and binomial coefficients. The obtained formulas rely on the application…
A Hadamard-Hitchcock decomposition of a multidimensional array is a decomposition that expresses the latter as a Hadamard product of several tensor rank decompositions. Such decompositions can encode probability distributions that arise…
We present a parametric family of Riordan arrays which are obtained by multiplying any Riordan array with a generalized Pascal array. In particular, we focus on some interesting properties of one-parameter Catalan triangles. We obtain…
Motivated by representation theory we exhibit an interior structure to Catalan sequences and many generalisations thereof. Certain of these coincide with well known (but heretofore isolated) structures. The remainder are new.
We give new identities for some symmetric polynomials. As applications of these identities, we obtain some formulas for a higher order analogue of Fibonacci and Lucas numbers.
We have found some patterns in some triangles.
In this paper we explore two types of tilings of a honeycomb strip and derive some closed form formulas for the number of tilings. Furthermore, we obtain some new identities involving tribonacci numbers, Padovan numbers and Narayana's cow…
Orthogonal array, a classical and effective tool for collecting data, has been flourished with its applications in modern computer experiments and engineering statistics. Driven by the wide use of computer experiments with both qualitative…
We combinatorially prove Tetranacci, Tetranacci-Fibonacci, and additional identities using only squares and dominoes on a hexagonal double-strip. Some of these are new proofs of old identities, and others we believe have never been seen…
In this paper, we connect two well established theories, the Fibonacci numbers and the Jordan algebras. We give a series of matrices, from literature, used to obtain recurrence relations of second-order and polynomial sequences. We also…
Matching astronomical catalogs in crowded regions of the sky is challenging both statistically and computationally due to the many possible alternative associations. Budav\'ari and Basu (2016) modeled the two-catalog situation as an…
Authentication of individuals via palmprint based biometric system is becoming very popular due to its reliability as it contains unique and stable features. In this paper, we present a novel approach for palmprint recognition and its…
Arc spaces have been introduced in algebraic geometry as a tool to study singularities but they show strong connections with combinatorics as well. Exploiting these relations we obtain a new approach to the classical Rogers-Ramanujan…
An elegant procedure which characterizes a decomposition of some class of binomial configurations into two other, resembling a definition of Pascal's Triangle, was given in \cite{gevay}. In essence, this construction was already presented…
Recently, a new generalization of Pascal's triangle, the so-called hyperbolic Pascal triangles were introduced. The mathematical background goes back to the regular mosaics in the hyperbolic plane. In this article, we investigate the paths…
In this article we introduce a new geometric object called hyperbolic Pascal simplex. This new object is presented by the regular hypercube mosaic in the 4-dimensional hyperbolic space. The definition of the hyperbolic Pascal simplex, whose…
A honeycomb array is an analogue of a Costas array in the hexagonal grid; they were first studied by Golomb and Taylor in 1984. A recent result of Blackburn, Etzion, Martin and Paterson has shown that (in contrast to the situation for…
We develop a unified framework for the study of properties involving diagonalizations of dense families in topological spaces. We provide complete classification of these properties. Our classification draws upon a large number of methods…
We provide elliptic extensions of elementary identities such as the sum of the first $n$ odd or even numbers, the geometric sum and the sum of the first $n$ cubes. Many such identities, and their $q$-analogues, are indefinite sums, and can…
The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers including those discovered recently by Paule and Schneider (2003).