Program to check if N is a Decagonal Number

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

Decagonal Number is a figurate number that extends the concept of triangular and square numbers to the decagon (10-sided polygon). The nth decagonal numbers count the number of dots in a pattern of n nested decagons, all sharing a common corner, where the ith decagon in the pattern has sides made of i dots spaced one unit apart from each other. The first few decagonal numbers are 1, 10, 27, 52, 85, 126, 175, …

Examples:



Input: N = 10
Output: Yes
Explanation:
Second decagonal number is 10.

Input: N = 30
Output: No

Approach:

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

    K^{th} Term = 4*K^{2} - 3*K

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

    => N =  4*K^{2} - 3*K
    => K = \frac{3 + \sqrt{16*N + 9}}{8}

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

Below is the implementation of the above approach:

C++

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// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to check if N is a
// Decagonal Number
bool isdecagonal(int N)
{
    float n
        = (3 + sqrt(16 * N + 9))
          / 8;
  
    // Condition to check if the
    // number is a decagonal number
    return (n - (int)n) == 0;
}
  
// Driver Code
int main()
{
    // Given Number
    int N = 10;
  
    // Function call
    if (isdecagonal(N)) {
        cout << "Yes";
    }
    else {
        cout << "No";
    }
    return 0;
}

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Java

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// Java program for the above approach 
import java.lang.Math;
  
class GFG{
      
// Function to check if N is a 
// decagonal number 
public static boolean isdecagonal(int N) 
    double n = (3 + Math.sqrt(16 * N + 9)) / 8
      
    // Condition to check if the 
    // number is a decagonal number 
    return (n - (int)n) == 0
  
// Driver code    
public static void main(String[] args)
{
          
    // Given number 
    int N = 10
      
    // Function call 
    if (isdecagonal(N))
    
        System.out.println("Yes");
    
    else 
    
        System.out.println("No");
    
}
}
  
// This code is contributed by divyeshrabadiya07    

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Python3

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# Python3 program for the above approach
import math
  
# Function to check if N is a
# decagonal number
def isdecagonal(N):
  
    n = (3 + math.sqrt(16 * N + 9)) / 8
      
    # Condition to check if the
    # number is a decagonal number
    return (n - int(n)) == 0
      
# Driver Code
if __name__=='__main__':
      
    # Given number
    N = 10
      
    # Function Call
    if isdecagonal(N):
        print('Yes')
    else:
        print('No')
  
# This code is contributed by rutvik_56

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C#

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// C# program for the above approach
using System;
  
class GFG{
      
// Function to check if N 
// is a decagonal Number
static bool isdecagonal(int N)
{
    double n = (3 + Math.Sqrt(16 * N + 9)) / 8;
      
    // Condition to check if the
    // number is a decagonal number
    return (n - (int)n) == 0;
}
      
// Driver Code
static public void Main ()
{
      
    // Given Number
    int N = 10;
      
    // Function call
    if (isdecagonal(N))
    {
        Console.Write("Yes");
    }
    else
    {
        Console.Write("No");
    }
}
}
  
// This code is contributed by ShubhamCoder

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Output:

Yes

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