Program to check if N is a Centered Decagonal Number

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

Centered Decagonal Number is centered figurative number that represents a decagon with dot in center and all other dot surrounding it in successive Decagonal Number form. The first few Centered decagonal numbers are 1, 11, 31, 61, 101, 151 …

Examples:

Input: N = 11
Output: Yes
Explanation:
Second Centered decagonal number is 11.

Input: N = 30
Output: No



Approach:

  1. The Kth term of the Centered Decagonal Number is given as

    K^{th} Term =  4*K^{2} - 4*K + 1

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

    => N =  {4*K^{2} - 4*K + 1}
    => K = \frac{5 + \sqrt{20*N + 5}}{10}

  3. If the value of K calculated using the above formula is an integer, then N is a Centered Decagonal Number.
  4. Else the number N is not a Centered 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 number N
// is a Centered decagonal number
bool isCentereddecagonal(int N)
{
    float n
        = (5 + sqrt(20 * N + 5))
          / 10;
  
    // Condition to check if N
    // is Centered Decagonal Number
    return (n - (int)n) == 0;
}
  
// Driver Code
int main()
{
    int N = 11;
  
    // Function call
    if (isCentereddecagonal(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 decagonal number or not 
import java.lang.Math;
  
class GFG{
      
// Function to check that the number 
// is a centered decagonal number 
public static boolean isCentereddecagonal(int N) 
    double n = (5 + Math.sqrt(20 * N + 5)) / 10
  
    // Condition to check if the number 
    // is a centered decagonal number 
    return (n - (int)n) == 0
  
// Driver Code
public static void main(String[] args) 
{
    int n = 11
  
    // Function call 
    if (isCentereddecagonal(n)) 
    
        System.out.println("Yes");
    
    else
    
        System.out.println("No");
    
}
}
  
// This code is contributed by ShubhamCoder

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Python3

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# Python3 program for the above approach
import numpy as np
  
# Function to check if the number N
# is a centered decagonal number
def isCentereddecagonal(N):
  
    n = (5 + np.sqrt(20 * N + 5)) / 10
  
    # Condition to check if N 
    # is centered decagonal number
    return (n - int(n)) == 0
  
# Driver Code 
N = 11
  
# Function call 
if (isCentereddecagonal(N)):
    print ("Yes"
else:
    print ("No")
  
# This code is contributed by PratikBasu

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C#

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// C# implementation to check that a number
// is a centered decagonal number or not 
using System;
  
class GFG{
      
// Function to check that the number 
// is a centered decagonal number 
static bool isCentereddecagonal(int N) 
    double n = (5 + Math.Sqrt(20 * N + 5)) / 10; 
      
    // Condition to check if the number 
    // is a centered decagonal number 
    return (n - (int)n) == 0; 
      
// Driver Code
static public void Main ()
{
    int n = 11; 
      
    // Function call 
    if (isCentereddecagonal(n)) 
    
        Console.Write("Yes");
    
    else
    
        Console.Write("No");
    
}
}
  
// This code is contributed by ShubhamCoder

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Output:

Yes

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