Program to check if N is a Octagonal Number

Given a number N, the task is to check if N is an Octagonal Number or not. If the number N is an Octagonal Number then print "Yes" else print "No".

Octagonal Number is the figure number that represent octagonal. Octagonal Numbers can be formed by placing triangular numbers on the four sides of a square. The first few Octagonal Numbers are 1, 8, 21, 40, 65, 96 ... 

Examples: 

Input: N = 8 
Output: Yes 
Explanation: 
Second octagonal number is 8.

Input: N = 30 
Output: No

Approach: 

1. The K^th term of the octagonal number is given as
K^{th} Term = 3*K^{2} - 2*K     
2. As we have to check whether the given number can be expressed as an Octagonal Number or not. This can be checked as: 

=> N = 3*K^{2} - 2*K     
=> K = \frac{2 + \sqrt{12*N + 4}}{6}     

3. If the value of K calculated using the above formula is an integer, then N is an Octagonal Number.
4. Else N is not an Octagonal Number.

Below is the implementation of the above approach:

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to check if N is a
// Octagonal Number
bool isoctagonal(int N)
{
    float n
        = (2 + sqrt(12 * N + 4))
          / 6;

    // Condition to check if the
    // number is a octagonal number
    return (n - (int)n) == 0;
}

// Driver Code
int main()
{
    // Given Number
    int N = 8;

    // Function call
    if (isoctagonal(N)) {
        cout << "Yes";
    }
    else {
        cout << "No";
    }
    return 0;
}

// Java program for the above approach
import java.io.*; 
import java.util.*; 

class GFG { 
    
// Function to check if N is a
// octagonal number
public static boolean isoctagonal(int N)
{
    double n = (2 + Math.sqrt(12 * N + 4)) / 6;
    
    // Condition to check if the
    // number is a octagonal number
    return (n - (int)n) == 0;
}
    
// Driver code 
public static void main(String[] args) 
{ 
    
    // Given Number
    int N = 8;
    
    // Function call
    if (isoctagonal(N))
    {
        System.out.println("Yes");
    }
    else
    {
        System.out.println("No");
    }
} 
} 

// This code is contributed by coder001

# Python3 program for the above approach 
from math import sqrt

# Function to check if N is a 
# octagonal number 
def isoctagonal(N):

    n = (2 + sqrt(12 * N + 4)) / 6; 

    # Condition to check if the 
    # number is a octagonal number 
    return (n - int(n)) == 0; 
    
# Driver Code 
if __name__ == "__main__": 

    # Given number 
    N = 8; 

    # Function call 
    if (isoctagonal(N)):
        print("Yes"); 
    
    else:
        print("No"); 

# This code is contributed by AnkitRai01

// C# program for the above approach
using System;

class GFG { 
    
// Function to check if N is a
// octagonal number
public static bool isoctagonal(int N)
{
    double n = (2 + Math.Sqrt(12 * N +
                              4)) / 6;
    
    // Condition to check if the
    // number is a octagonal number
    return (n - (int)n) == 0;
}
    
// Driver code 
public static void Main(String[] args) 
{ 
    
    // Given number
    int N = 8;
    
    // Function call
    if (isoctagonal(N))
    {
        Console.WriteLine("Yes");
    }
    else
    {
        Console.WriteLine("No");
    }
} 
} 

// This code is contributed by Rohit_ranjan

<script>

// JavaScript program for the above approach

// Function to check if N is a
// Octagonal Number
function isoctagonal(N)
{
    var n = (2 + Math.sqrt(12 * N + 4)) / 6;

    // Condition to check if the
    // number is a octagonal number
    return (n - parseInt(n) == 0);
}

// Given Number
var N = 8;

// Function call
if (isoctagonal(N)) {
    document.write("Yes");
}
else {
    document.write("No");
}

</script>

Yes

Time Complexity: O(log N) as sqrt function is being used
Auxiliary Space: O(1)