n’th Pentagonal Number

Given an integer n, find the nth Pentagonal number. First three pentagonal numbers are 1, 5 and 12 (Please see below diagram).

The n’th pentagonal number P n is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex [Source Wiki ]

Examples:

Input: n = 1
Output: 1
Input: n = 2
Output: 5
Input: n = 3
Output: 12

In general, a polygonal number (triangular number, square number, etc) is a number represented as dots or pebbles arranged in the shape of a regular polygon. The first few pentagonal numbers are:

If s is the number of sides in a polygon, the formula for the nth s-gonal number P (s, n) is

nth s-gonal number P(s, n) = (s - 2)n(n-1)/2 + n
If we put s = 5, we get
n'th Pentagonal number Pn = 3*n*(n-1)/2 + n

Examples:

Triangular number

Square number

Pentagonal Number

Below are the implementations of above idea in different programming languages.

C/C++ // C program for above approach
#include <stdio.h>
#include <stdlib.h>
// Finding the nth Pentagonal Number
int pentagonalNum(int n)
{
return (3*n*n - n)/2;
}
// Driver program to test above function
int main()
{
int n = 10;
printf("10th Pentagonal Number is = %d \n \n",
pentagonalNum(n));
return 0;
} Java // Java program for above approach
class Pentagonal
{
int pentagonalNum(int n)
{
return (3*n*n - n)/2;
}
}
public class GeeksCode
{
public static void main(String[] args)
{
Pentagonal obj = new Pentagonal();
int n = 10;
System.out.printf("10th petagonal number is = "
+ obj.pentagonalNum(n));
}
} python # Python program for finding pentagonal numbers
def pentagonalNum( n ):
return (3*n*n - n)/2
#Script Begins
n = 10
print "10th Pentagonal Number is = ", pentagonalNum(n)
#Scripts Ends

Output:

10th Pentagonal Number is = 145

Reference:

https://en.wikipedia.org/wiki/Polygonal_number

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