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n’th Pentagonal Number

  • Difficulty Level : Medium
  • Last Updated : 16 Jul, 2021

Given an integer n, find the nth Pentagonal number. The 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: 
 

Pentagonal Number



 

Pentagonal Number

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

C++




// C++ program for above approach
#include<bits/stdc++.h>
using namespace std;
 
// Finding the nth pentagonal number
int pentagonalNum(int n)
{
    return (3 * n * n - n) / 2;
}
 
// Driver code
int main()
{
    int n = 10;
     
    cout << "10th Pentagonal Number is = "
         << pentagonalNum(n);
 
    return 0;
}
 
// This code is contributed by Code_Mech

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

C#




// C# program for above approach
using System;
 
class GFG {
     
    static int pentagonalNum(int n)
    {
        return (3 * n * n - n) / 2;
    }
 
    public static void Main()
    {
        int n = 10;
         
        Console.WriteLine("10th petagonal"
        + " number is = " + pentagonalNum(n));
    }
}
 
// This code is contributed by vt_m.

PHP




<?php
// PHP program for above approach
 
// Finding the nth Pentagonal Number
function pentagonalNum($n)
{
    return (3 * $n * $n - $n) / 2;
}
 
// Driver Code
$n = 10;
echo "10th Pentagonal Number is = ",
                  pentagonalNum($n);
 
// This code is contributed by ajit
?>

Javascript




<script>
 
// Javascript program for above approach
 
    function pentagonalNum(n)
    {
        return (3 * n * n - n) / 2;
    }
 
// Driver code to test above methods
 
        let n = 10;
           
        document.write("10th petagonal"
        + " number is = " + pentagonalNum(n));
          
         // This code is contributed by avijitmondal1998.
</script>

Output : 
 

10th Pentagonal Number is = 145

Time Complexity: O(1)
Auxiliary Space: O(1)
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 write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 

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