Program to check if N is a Centered nonadecagonal number

Given an integer N, the task is to check if it is a Centered nonadecagonal number or not.

Centered nonadecagonal number represents a dot in the centre and other dots surrounding it in successive nonadecagon(19 sided polygon) layers.The first few Centered nonadecagonal numbers are 1, 20, 58, 115, 191, 286

Examples:

Input: N = 20
Output: Yes
Explanation:
20 is the Second Centered nonadecagonal number is 20.

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



Approach: To solve the problem mentioned above we have to know that the Kth term of the Centered nonadecagonal number is given as: K^{th} Term = \frac {19*N^{2} - 19*N + 2}{2}

As we have to check that the given number can be expressed as a Centered nonadecagonal number or not. This can be checked by generalizing the formula to:

=> N =  \frac {19 * k ^ {2} - 19 * k + 2} {2}

=> K = \frac{19 + \sqrt{152 * N + 209}} {38}

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

Below is the implementation of the above approach:

C++

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// C++ implementation to check that
// a number is a Centered
// nonadecagonal number or not
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to check that the
// number is a Centered
// nonadecagonal number
bool isCenterednonadecagonal(int N)
{
    float n = (19
               + sqrt(152 * N + 209))
              / 38;
  
    // Condition to check if the
    // number is a Centered
    // nonadecagonal number
    return (n - (int)n) == 0;
}
  
// Driver Code
int main()
{
    int n = 20;
  
    // Function call
    if (isCenterednonadecagonal(n)) {
        cout << "Yes";
    }
    else {
        cout << "No";
    }
    return 0;
}

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Java

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// Java implementation to check that
// a number is a Centered
// nonadecagonal number or not
class GFG{
  
// Function to check that the
// number is a Centered
// nonadecagonal number
static boolean isCenterednonadecagonal(int N)
{
    float n = (float)((19 + Math.sqrt(152 * N + 
                                      209)) / 38);
  
    // Condition to check if the
    // number is a Centered
    // nonadecagonal number
    return (n - (int)n) == 0;
}
  
// Driver Code
public static void main(String[] args)
{
    int n = 20;
  
    // Function call
    if (isCenterednonadecagonal(n))
    {
        System.out.print("Yes");
    }
    else 
    {
        System.out.print("No");
    }
}
}
  
// This code is contributed by sapnasingh4991

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Python3

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# Python3 implementation to check that
# a number is a Centered
# nonadecagonal number or not
import math
  
# Function to check that the
# number is a Centered
# nonadecagonal number
def isCenterednonadecagonal(N):
  
    n = (19 + math.sqrt(152 * N + 
                        209)) / 38;
  
    # Condition to check if the
    # number is a Centered
    # nonadecagonal number
    return (n - int(n)) == 0;
  
# Driver Code
n = 20;
  
# Function call
if (isCenterednonadecagonal(n)):
    print("Yes");
  
else:
    print("No");
  
# This code is contributed by Code_Mech

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C#

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// C# implementation to check that
// a number is a Centered
// nonadecagonal number or not
using System;
class GFG{
  
// Function to check that the
// number is a Centered
// nonadecagonal number
static bool isCenterednonadecagonal(int N)
{
    float n = (float)((19 + Math.Sqrt(152 * N + 
                                      209)) / 38);
  
    // Condition to check if the
    // number is a Centered
    // nonadecagonal number
    return (n - (int)n) == 0;
}
  
// Driver Code
public static void Main()
{
    int n = 20;
  
    // Function call
    if (isCenterednonadecagonal(n))
    {
        Console.Write("Yes");
    }
    else
    {
        Console.Write("No");
    }
}
}
  
// This code is contributed by Nidhi_biet

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Output:

Yes

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