Program to check if N is a Centered dodecagonal number

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

Centered Dodecagonal Number represents a dot in the center and other dots surrounding it in successive Dodecagonal Number(12 sided polygon) layers. The first few Centered Dodecagonal Numbers are 1, 13, 37, 73 ... 

Examples:  

Input: N = 13 
Output: Yes 
Explanation: 
Second Centered dodecagonal number is 13.

Input: N = 30 
Output: No 

Approach:  

1 The K^th term of the Centered Dodecagonal Number is given as:
K^{th} Term = 6*K^{2} - 6*K + 1      
 

2. As we have to check that the given number can be expressed as a Centered Dodecagonal Number or not. This can be checked as: 

=> N = {6*K^{2} - 6*K + 1}      
=> K = \frac{6 + \sqrt{24*N + 12}}{12}       

3. If the value of K calculated using the above formula is an integer, then N is a Centered Dodecagonal Number.

4. Else the number N is not a Centered Dodecagonal 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 number N
// is a Centered dodecagonal number
bool isCentereddodecagonal(int N)
{
    float n
        = (6 + sqrt(24 * N + 12))
          / 12;

    // Condition to check if N
    // is a Centered Dodecagonal Number
    return (n - (int)n) == 0;
}

// Driver Code
int main()
{
    // Given Number
    int N = 13;

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

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

class GFG{

// Function to check if number N
// is a centered dodecagonal number
static boolean isCentereddodecagonal(int N)
{
    float n = (float) ((6 + Math.sqrt(24 * N + 
                                      12)) / 12);

    // Condition to check if N is a 
    // centered dodecagonal number
    return (n - (int)n) == 0;
}

// Driver Code
public static void main(String[] args)
{
    
    // Given Number
    int N = 13;

    // Function call
    if (isCentereddodecagonal(N))
    {
        System.out.print("Yes");
    }
    else
    {
        System.out.print("No");
    }
}
}

// This code is contributed by sapnasingh4991

# Python3 program for the above approach
import numpy as np

# Function to check if the number N
# is a centered dodecagonal number
def isCentereddodecagonal(N):

    n = (6 + np.sqrt(24 * N + 12)) / 12

    # Condition to check if N 
    # is a centered dodecagonal number
    return (n - int(n)) == 0

# Driver Code 
N = 13

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

# This code is contributed by PratikBasu

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

class GFG{

// Function to check if number N
// is a centered dodecagonal number
static bool isCentereddodecagonal(int N)
{
    float n = (float) ((6 + Math.Sqrt(24 * N + 
                                      12)) / 12);

    // Condition to check if N is a 
    // centered dodecagonal number
    return (n - (int)n) == 0;
}

// Driver Code
public static void Main(string[] args)
{
    
    // Given Number
    int N = 13;

    // Function call
    if (isCentereddodecagonal(N))
    {
        Console.Write("Yes");
    }
    else
    {
        Console.Write("No");
    }
}
}

// This code is contributed by rutvik_56

<script>
// Javascript program for the above approach

// Function to check if number N
// is a Centered dodecagonal number
function isCentereddodecagonal(N)
{
    let n
        = (6 + Math.sqrt(24 * N + 12))
          / 12;

    // Condition to check if N
    // is a Centered Dodecagonal Number
    return (n - parseInt(n)) == 0;
}

// Driver Code
// Given Number
let N = 13;

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

// This code is contributed by subham348.
</script>

Yes

Time Complexity: O(logN) because it is using inbuilt sqrt function

Auxiliary Space: O(1)