Polygonal Number Formula
A polygonal number is a type of figurate number that is a generalization of triangular, square, etc., to an-gon for an arbitrary positive integer. The above diagrams graphically illustrate the process by which the polygonal numbers are built up. Starting with the th triangular number, then. We use these relationships to derive the general formula for polygonal numbers. In finding the number of dots in a polygon, two quantities vary. The first varying quantity is the number of sides of the polygon which we have denoted above as n. The second quantity is the size of the polygon which we denote as k. The General formula to find the sum of first n K th Dimensional S sided polygonal numbers and a www.iosrjournals.org 6 | Page 4.4 5D Polygonal Numbers (5 Dimensional Polygonal Numbers) 5D Triangular Numbers1,6,21,56,126,252,462... The General formula to find the sum of first n K th Dimensional S sided polygonal numbers and a www.iosrjournals.org 9 | Pageσ m n 5 =n n + 1 n + 2 n + 3 …. n + mm+ 5 ! x (120n 4 + 240mn 3 −150m 2 n 2 −90mn 2 −90m 2 n + 30m 3 n + 4m + 11m 2 −16m 3 + m) VIII. Figures d IX. In mathematics, a polygonal number is a number represented as dots or pebbles arrayed in the shape of a polygon. The dots were thought of as alphas (units). These are one type of figurate numbers. The number 10, for example, can be arranged as a triangle (see triangular number): If s is the number of sides in a polygon, the formula for the n th s-gonal number P (s, n) is. or. The n th s-gonal number is also related to the triangular numbers T n as follows: Thus: If s is the number of sides in a polygon, the formula for the n th s-gonal number P (s, n) is. or. The n th s-gonal number is also related to the triangular numbers T n as follows: Thus: For a given s-gonal number P (s, n) = x, one can find n by. A previous post suggested an approach for generating a formula for higher-dimensional polygonal numbers. Future posts will elaborate on these numbers some more. Some examples of two-dimensional polygonal numbers are: the triangular numbers: 1, 3, 6, 10, 15, ... Similarly, the first few 3-highly polygonal numbers up to are 1, 15, 36, 45, 325, 561, 1225, 1540, 3025, ... (OEIS A062712). There are no 4-highly polygonal numbers of this type less than except for 1. The generating function for the-gonal numbers is given by the beautiful formula. The first 6 values in the column Sum of Reciprocals, for triangular to octagonal numbers, come from a published solution to the general problem, which also gives a general formula for any number of sides, in terms of the digamma function.
 In finding the number of dots in a polygon, two quantities vary. The first varying quantity is the number of sides of the polygon which we have denoted above as n. The second quantity is the size of the polygon which we denote as k.