The fascinating mathematical beauty of triangular numbers (original) (raw)
Related papers
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.
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.