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

Last Updated : 28 Nov, 2022

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 center 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 K^th 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++

// 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; }

Java

// 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

Python3

# 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

// 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

JavaScript

<script> // javascript program for the above approach  // Function to check if number N // is a centered pentagonal number function isCenteredpentagonal(N) {     var n =  ((5 + Math.sqrt(40 * N -                                        15)) / 10);      // Condition to check if N is a     // centered pentagonal number     return (n - parseInt(n) == 0); }  // Driver Code //Given Number var N = 6;  // Function call if (isCenteredpentagonal(N)) {     document.write("Yes"); } else  {     document.write("No"); }  // This code is contributed by Amit Katiyar  </script>