n’th Pentagonal Number

3.6

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 Pn 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: 1, 5, 12, etc.
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

Pentagonal Number

Square number

Pentagonal Number

Pentagonal 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

This article is contributed by Mazhar Imam Khan. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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