Program to check if N is a Centered Octadecagonal number

Given an integer N, the task is to check if it is a Centered Octadecagonal number or not. Print “yes” if it is otherwise output is “no”.

Centered Octadecagonal number represents a dot in the centre and others dot are arranged around it in successive layers of octadecagon(18 sided polygon). The first few Centered Octadecagonal numbers are 1, 19, 55, 109, 181, 271, 379, …

Examples:

Input: N = 19 
Output: Yes 
Explanation:
19 is the Second Centered Octadecagonal number is 19.

Input: 38 
Output: No 
Explanation:
38 is not a Centered Octadecagonal number. 

Approach: To solve the problem mentioned above we know that the K^th term of the Centered Octadecagonal number is given as: 
As we have to check that the given number can be expressed as a Centered Octadecagonal number or not. This can be checked by generalizing the equation as: 

=> 
=> 

Finally, check the value of computation using this formula if it is an integer, then it means that N is a Centered Octadecagonal number.

Below is the implementation of the above approach: 

C++

// C++ implementation to check that
// a number is a Centered
// Octadecagonal number or not

#include <bits/stdc++.h>
using namespace std;

// Function to check that the
// number is a Centered
// Octadecagonal number
bool isCenteredOctadecagonal(int N)
{

    // Implement the formula generated
    float n = (9 + sqrt(36 * N + 45)) / 18;

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

// Driver Code
int main()
{
    int n = 19;

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

Java

// Java implementation to check that
// a number is a centered
// octadecagonal number or not
import java.lang.Math;

class GFG{

// Function to check that the
// number is a centered
// octadecagonal number
public static boolean isCenteredOctadecagonal(int N)
{

    // Implement the formula generated
    double n = (9 + Math.sqrt(36 * N + 45)) / 18;

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

// Driver Code
public static void main(String[] args)
{
    int n = 19;

    // Function call
    if (isCenteredOctadecagonal(n))
    {
        System.out.println("Yes");
    }
    else
    {
        System.out.println("No");
    }
}
}

// This code is contributed by divyeshrabadiya07

Python3

# Python3 implementation to check that
# a number is a centered octadecagonal
# number or not
import math

# Function to check that the
# number is a centered
# octadecagonal number
def isCenteredOctadecagonal(N):

    # Implement the formula generated
    n = (9 + math.sqrt(36 * N + 45)) / 18;

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

# Driver code
if __name__=='__main__':

    n = 19

    # Function call
    if isCenteredOctadecagonal(n):
        print('Yes')
    else:
        print('No')

# This code is contributed by rutvik_56

C#

// C# implementation to check that
// a number is a centered
// octadecagonal number or not
using System;

class GFG{

// Function to check that the
// number is a centered
// octadecagonal number
static bool isCenteredOctadecagonal(int N)
{

    // Implement the formula generated
    double n = (9 + Math.Sqrt(36 * N + 45)) / 18;

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

// Driver Code
static public void Main ()
{
    int n = 19;

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

//This code is contributed by ShubhamCoder

Javascript

<script>

// Javascript implementation to check that
// a number is a Centered Octadecagonal
// number or not

// Function to check that the
// number is a Centered
// Octadecagonal number
function isCenteredOctadecagonal(N)
{

    // Implement the formula generated
    let n = parseInt((9 + Math.sqrt(36 * N + 45)) / 18);

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

// Driver Code
let n = 19;

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

// This code is contributed by souravmahato348

</script>

Output:

Yes

Time Complexity: O(logn) for given n, as it is using inbuilt sqrt function
Auxiliary Space: O(1)