Pattern Classification of Continued Fractions With Square Number as Base

Abstract

In number theory, study of number sequences is an enthusiastic area. Among these the sequence of polygonal numbers gives a unique richness in is applicability. Polygonal numbers which have both order, rank of are of various dimensions. Here, the study is only on two dimensional figurate numbers. Here the ratios of polygonal numbers are studied as continued fractions. In this paper different types of properties and characteristics of sequence of continued fraction which represent ratios of polygonal number of same rank are discussed. As any polygonal can be sub divided into smaller squares, square numbers have been taken as base here. Based upon on this study the nature and periodicity of the general sequence can be studied in detail.Notations:1. 〈 u0, u1, u2, u3, …., un〉 – Continued fraction expansion2. S5,n - Pentagonal number3. S6,n - Hexagonal number4. S7,n - Heptagonal number AMS Classification: 11A55