On Square-Triangular Numbers

Abstract

INTRODUCTION

In mathematics and physics, there are lots of integer sequences, which are used in almost every field of modern sciences. Admittedly, the Fibonacci, Pell and Pell-Lucas sequence are some of the most famous and curious numerical sequences in mathematics and have been widely studied from both algebraic and combinatorial perspective. Also, there is the Square-Triangular sequence, which is as important as that sequence. Square-Triangular sequence {STn} is defined by recurrence 34STn-1-STn-2 + 2, n ≥ 2 with ST1 = 1 and ST2= 36. The main objective of this study is to discuss properties, relationships, and applications of Square-Triangular sequence.

METHODS

This study utilized the expository and descriptive methods of research. Descriptive research was used to describe and discuss in detail the concepts related to Square-Triangular Numbers. In this study, the theorems involving generators, properties, and relationships of Square-Triangular numbers were proven, presented and discussed comprehensively. All data and information were described and verify through illustrations or examples.

RESULTS

There are three ways to determine the Square-triangular numbers such as the recurrence relation, the Binet's formula, and Pell's Equation, solutions to this equation produce Square-triangular number. This number has captivating relationships to other forms of numbers such as Pell numbers, Pell-Lucas numbers. The ratio of consecutive Square-Triangular numbers is approaching to the silver ratio as n get large. These numbers have properties showing significant results when its terms are added and interesting relationships when its terms are manipulated. These properties were proven using the algebraic method, limits and mathematical induction. These numbers can also generate Pythagorean triples. The area of a Square-Triangular right triangle and the solution of the Diophantine equation can also be computed using the Square-triangular number n. These numbers have application in the Cryptography.

DISCUSSIONS

The square-triangular sequence has three generators such as recurrence relation, Binet's formula, and Pell's equation. Adding or multiplying the terms of Square-triangular numbers results in interesting properties that produced certain patterns. The square-triangular number revealed unusual relationships to other forms of numbers such as Pell number, Pell-Lucas number, Silver ratio, Pythagorean triples, and Diophantine equation. These numbers had useful application in cryptography.