Program to check if N is a Icosihenagonal number

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

Icosihenagonal number is class of figurate number. It has 21 – sided polygon called Icosihenagon. The n-th Icosihenagonal number counts the 21 number of dots and all others dots are surrounding with a common sharing corner and make a pattern. The first few Icosihenagonal numbers are 1, 21, 60, 118, 195, 291, 406…

Examples:

Input: N = 21
Output: Yes
Explanation:
Second icosihenagonal number is 21.

Input: N = 30
Output: No



Approach:

  1. The Kth term of the icosihenagonal number is given as

    K^{th} Term =  \frac{19*K^{2} - 17*K}{2}

  2. As we have to check that the given number can be expressed as a icosihenagonal number or not. This can be checked as follows –

    => N =  \frac{19*K^{2} - 17*K}{2}
    => K = \frac{17 + \sqrt{152*N + 289}}{38}

  3. Finally, check the value of computed using this formulae is an integer, which means that N is a icosihenagonal number.

Below is the implementation of the above approach:

C++

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// C++ implementation to check that
// a number is icosihenagonal number or not
  
#include <bits/stdc++.h>
  
using namespace std;
  
// Function to check that the
// number is a icosihenagonal number
bool isicosihenagonal(int N)
{
    float n
        = (17 + sqrt(152 * N + 289))
          / 38;
  
    // Condition to check if the
    // number is a icosihenagonal number
    return (n - (int)n) == 0;
}
  
// Driver Code
int main()
{
    int i = 21;
  
    // Function call
    if (isicosihenagonal(i)) {
        cout << "Yes";
    }
    else {
        cout << "No";
    }
    return 0;
}

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Java

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// Java implementation to check that a 
// number is icosihenagonal number or not
class GFG{
  
// Function to check that the number
// is a icosihenagonal number
static boolean isicosihenagonal(int N)
{
    float n = (float) ((17 + Math.sqrt(152 * N + 
                                       289)) / 38);
  
    // Condition to check if the number
    // is a icosihenagonal number
    return (n - (int)n) == 0;
}
  
// Driver Code
public static void main(String[] args)
{
    int i = 21;
  
    // Function call
    if (isicosihenagonal(i))
    {
        System.out.print("Yes");
    }
    else
    {
        System.out.print("No");
    }
}
}
  
// This code is contributed by 29AjayKumar

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Python3

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# Python3 implementation to check that 
# a number is icosihenagonal number or not 
import math 
  
# Function to check that the number
# is a icosihenagonal number 
def isicosihenagonal(N): 
  
    n = (17 + math.sqrt(152 * N + 289)) / 38
  
    # Condition to check if the number
    # is a icosihenagonal number 
    return (n - int(n)) == 0
  
# Driver Code 
i = 21
  
# Function call 
if isicosihenagonal(i): 
    print("Yes")
else
    print("No")
      
# This code is contributed by divyamohan123

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C#

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// C# implementation to check that a 
// number is icosihenagonal number or not
using System;
  
class GFG{
  
// Function to check that the number
// is a icosihenagonal number
static bool isicosihenagonal(int N)
{
    float n = (float)((17 + Math.Sqrt(152 * N + 
                                      289)) / 38);
  
    // Condition to check if the number
    // is a icosihenagonal number
    return (n - (int)n) == 0;
}
  
// Driver Code
public static void Main()
{
    int i = 21;
  
    // Function call
    if (isicosihenagonal(i))
    {
        Console.Write("Yes");
    }
    else
    {
        Console.Write("No");
    }
}
}
  
// This code is contributed by Code_Mech

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Output:

Yes

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