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.
Generalized Factoriangular Numbers and Factoriangular Triangles
A factoriangular number is defined as the sum of corresponding factorial and triangular number. This paper aims to generalize this number as sum of any factorial and any triangular number and explore such generalization. This study is a basic research in number theory that uses mathematical exposition and exploration. The generalized factoriangular number is of the form ! k nT , where ! n is the factorial of a natural number n and k T is the th k triangular number. When nk , the sum is an ordinary factoriangular number. A consequence of the generalization is the creation of interesting Pascal-like triangles that are hereby called factoriangular triangles and formation of their corresponding integer sequences. Generalized factoriangular numbers and factoriangular triangles can be utilized as recreational mathematics for students. Further generalizations of factoriangular number and expositions on factoriangular triangles can be done next.
On the Sum of Corresponding Factorials and Triangular Numbers: Runsums, Trapezoids and Politeness
When corresponding numbers in the sequence of factorials and sequence of triangular numbers are added, a new sequence of natural numbers is formed. In this study, these positive integers are called factoriangular numbers. Closely related to these new numbers are the runsums, trapezoidal and polite numbers. Some theorems on runsum representations of factoriangular numbers are proven here, as well as, a theorem on factoriangular number being represented as difference of two triangular numbers. Unambiguous definitions of trapezoidal number and number of trapezoidal arrangements are also given, including how these differ from runsum, polite number, number of runsums and politeness.
A note on number triangles that are almost their own production matrix
arXiv: Combinatorics, 2018
We characterize a family of number triangles whose production matrices are closely related to the original number triangle. We study a number of such triangles that are of combinatorial significance. For a specific subfamily, these triangles relate to sequences that have interesting convolution recurrences and continued fraction generating functions.
A note on Sierpi '{n}ski problem related to triangular numbers
2008
In this note we show that the system of equations t_{x}+t_{y}=t_{p},\quad t_{y}+t_{z}=t_{q},\quad t_{x}+t_{z}=t_{r}, where tx=x(x+1)/2t_{x}=x(x+1)/2tx=x(x+1)/2 is a triangular number, has infinitely many solutions in integers. Moreover we show that this system has rational three-parametric solution. Using this result we show that the system t_{x}+t_{y}=t_{p},\quad t_{y}+t_{z}=t_{q},\quad t_{x}+t_{z}=t_{r},\quad t_{x}+t_{y}+t_{z}=t_{s} has infinitely many rational two-parametric solutions.
Arithmetic and Polygonal Properties of Number Triangle
2022
Among several number triangles that exist in mathematics, Pascal's triangle is the most fascinating and prominent structure possessing numerous properties. In this paper, we will introduce a number triangle consisting of positive integers arranged in ascending to descending order so that the entries in each row would be mirror images with respect to its middle number. Using this simple number triangle, we have proved six amusing results which interestingly depend only as functions of its centred numbers.Moreover some of the concepts and results discussed in this paper have connection with Tamil literary work and triangular numbers.
Some Facinating Properties of Balancing Numbers
The study of number sequences has been a source of attraction to the mathematicians since ancient times. Since then many of them are focusing their interest on the study of the fascinating triangular numbers. In a recent study Behera and Panda tried to find the solutions of the Diophantine equation 1+2+ and found that the square of any n ∈ℤ + satisfying this equation is a triangular number. It can be also shown that if r ∈ℤ + satisfies the above equation then r r + 2 is also a triangular number. If a pair (n, r) constitutes a solution of the above equation then n is called a balancing number and r is called the balancer corresponding to n. In the joint paper "On the square roots of triangular numbers" published in "The Fibonacci Quarterly" in 1999, Behera and Panda introduced balancing numbers and studied many important properties of these numbers. In this paper we establish some other interesting arithmetic-type, de-Moivre's-type and trigonometric-type properties of balancing numbers. We also establish a most important property concerning the greatest common divisor of two balancing numbers.
Triangular Numbers and Elliptic Curves
Rocky Mountain Journal of Mathematics, 1996
Some arithmetic of elliptic curves and theory of elliptic surfaces is used to find all rational solutions (r, s, t) in the function field Q(m, n) of the pair of equations r(r + 1)/2 = ms(s + 1)/2 r(r + 1)/2 = nt(t + 1)/2.
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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