The fascinating mathematical beauty of triangular numbers (original) (raw)
On Triangular and Trapezoidal Numbers
Asia Pacific Journal of Multidisciplinary Research, 2015
The nth triangular number, denoted by T (n), is defined as the sum of the first consecutive positive integers. It can be represented in the form of a triangular grid of points. As a result, this study discusses the triangular numbers and its properties. To distinguish whether a positive integer N is a triangular number or not, it has to comply with 8N + 1 which is a perfect square; characteristics of odd and even triangular numbers; sum of two consecutive triangular numbers with same parity i.e., the formula for 2
Triangular Numbers and Their Inherent Properties
Variant Construction from Theoretical Foundation to Applications, 2018
A method to classify one-dimensional binary sequences using three parameters intrinsic to the sequence itself is introduced. The classification scheme creates combinatorial patterns that can be arranged in a two-dimensional triangular structure. Projections of this structure contain interesting properties related to the Pascal triangle numbers. The arrangement of numbers within the triangular structure has been named "triangular numbers", and the essential parameters, elementary equation, and sequencing schemes are discussed as well as visualizations of sample distributions, special cases, and search results. We believe this to be a novel finding as sequences generated using this method are not contained in the On-Line Encyclopedia of Integer Sequences or OEIS.
On the representation of integers as sums of triangular numbers
Aequationes Mathematicae, 1995
In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study reveals several interesting results. If n ≥ 0 is a non-negative integer, then the n th triangular number is Tn = n(n+1) 2 . Let k be a positive integer. We denote by δ k (n) the number of representations of n as a sum of k triangular numbers. Here we use the theory of modular forms to calculate δ k (n). The case where k = 24 is particularly interesting. It turns out that if n ≥ 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 2 12 (2 12 -1)δ 24 (n -3). Furthermore the formula for δ 24 (n) involves the Ramanujan τ (n)-function. As a consequence, we get elementary congruences for τ (n). In a similar vein, when p is a prime we demonstrate δ 24 (p k -3) as a Dirichlet convolution of σ 11 (n) and τ (n). It is also of interest to know that this study produces formulas for the number of lattice points inside k-dimensional spheres.
SlideShare, 2021
Throughout mathematics history, mathematicians have created a triangular array of numbers. Famous among these number triangles is Pascal’s Triangle which has marked its prominence in many areas of mathematics and even extends its usefulness in the sciences. This paper presents an inventory of number triangles known and recognized in the mathematics world and takes a look at the newly-found triangular array of numbers generated by a function and its link to Pascal’s Triangle, particularly to the Tetrahedral Numbers.
On the Sum of Corresponding Factorials and Triangular Numbers: Some Preliminary Results
A new sequence of natural numbers can be formed by adding corresponding factorials and triangular numbers. In this paper, such numbers were named factoriangular numbers. Mathematical experimentations on these numbers resulted to the establishment of some of its characteristics. These include the parity, compositeness, the number and sum of its positive divisors, abundancy and deficiency, Zeckendorf’s decomposition, end digits, and digital roots of factoriangular numbers. Several theorems and corollaries were proven and some conjectures were also presented.
A note on a one-parameter family of non-symmetric number triangles
Opuscula Mathematica, 2012
The recently growing interest in special Clifford Algebra valued polynomial solutions of generalized Cauchy-Riemann systems in (n + 1)-dimensional Euclidean spaces suggested a detailed study of the arithmetical properties of their coefficients, due to their combinatoric relevance. This concerns, in particular, a generalized Appell sequence of homogeneous polynomials whose coefficient's set can be treated as a one-parameter family of non-symmetric triangles of fractions. The discussion of its properties, similar to those of the ordinary Pascal triangle (which itself does not belong to the family), is carried out in this paper.
Triangular recurrences, generalized Eulerian numbers, and related number triangles
Advances in Applied Mathematics, 2023
Many combinatorial and other number triangles are solutions of recurrences of the Graham-Knuth-Patashnik (GKP) type. Such triangles and their defining recurrences are investigated analytically. They are acted upon by a transformation group generated by two involutions: a left-right reflection and an upper binomial transformation, acting row-wise. The group also acts on the bivariate exponential generating function (EGF) of the triangle. By the method of characteristics, the EGF of any GKP triangle has an implicit representation in terms of the Gauss hypergeometric function. There are several parametric cases when this EGF can be obtained in closed form. One is when the triangle elements are the generalized Stirling numbers of Hsu and Shiue. Another is when they are generalized Eulerian numbers of a newly defined kind. These numbers are related to the Hsu-Shiue ones by an upper binomial transformation, and can be viewed as coefficients of connection between polynomial bases, in a manner that generalizes the classical Worpitzky identity. Many identities involving these generalized Eulerian numbers and related generalized Narayana numbers are derived, including closed-form evaluations in combinatorially significant cases.
Number Theory: New York Seminar 1991–1995
1996
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