n'th Pentagonal Number - GeeksforGeeks

n’th Pentagonal Number

Difficulty Level : Medium
Last Updated : 31 Mar, 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 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: 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 P_n = 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

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