Program to check if N is a Centered Pentagonal Number or not

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

Centered Pentagonal Number is a centered figurate number that represents a pentagon with a dot in the centre and other dots surrounding it in pentagonal layers successively. The first few Centered Pentagonal Number are 1, 6, 16, 31, 51, 76, 106 …

Examples:

Input: N = 6
Output: Yes
Explanation:
Second Centered pentagonal number is 6.

Input: N = 20
Output: No



Approach:

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

    K^{th} Term =  \frac{5*K^{2} - 5*K + 2}{2}

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

    => N =  \frac{5*K^{2} - 5*K + 2}{2}
    => K = \frac{5 + \sqrt{40*N - 15}}{10}

  3. If the value of K calculated using the above formula is an integer, then N is a Centered Pentagonal Number.
  4. Else the number N is not a Centered Pentagonal 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 pentagonal number
bool isCenteredpentagonal(int N)
{
    float n
        = (5 + sqrt(40 * N - 15))
          / 10;
  
    // Condition to check if N is a
    // Centered pentagonal number
    return (n - (int)n) == 0;
}
  
// Driver Code
int main()
{
    // Given Number
    int N = 6;
  
    // Function call
    if (isCenteredpentagonal(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.util.*;
  
class GFG{
  
// Function to check if number N
// is a centered pentagonal number
static boolean isCenteredpentagonal(int N)
{
    float n = (float) ((5 + Math.sqrt(40 * N - 
                                      15)) / 10);
  
    // Condition to check if N is a
    // centered pentagonal number
    return (n - (int)n) == 0;
}
  
// Driver Code
public static void main(String[] args)
{
      
    // Given Number
    int N = 6;
  
    // Function call
    if (isCenteredpentagonal(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 program for the above approach 
import numpy as np
  
# Function to check if number N 
# is a centered pentagonal number 
def isCenteredpentagonal(N):
  
    n = (5 + np.sqrt(40 * N - 15)) / 10
  
    # Condition to check if N is a 
    # centered pentagonal number 
    return (n - int(n)) == 0
  
# Driver Code 
N = 6
  
# Function call 
if (isCenteredpentagonal(N)):
    print ("Yes")
else
    print ("No"
  
# This code is contributed by PratikBasu

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C#

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// C# program for the above approach
using System;
  
class GFG{
  
// Function to check if number N
// is a centered pentagonal number
static bool isCenteredpentagonal(int N)
{
    float n = (float) ((5 + Math.Sqrt(40 * N - 
                                      15)) / 10);
  
    // Condition to check if N is a
    // centered pentagonal number
    return (n - (int)n) == 0;
}
  
// Driver Code
public static void Main(string[] args)
{
      
    // Given number
    int N = 6;
  
    // Function call
    if (isCenteredpentagonal(N))
    {
        Console.Write("Yes");
    }
    else
    {
        Console.Write("No");
    }
}
}
  
// This code is contributed by rutvik_56

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Output:

Yes

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