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 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

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:

=> 
=> 

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#

// 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>

Output:

Yes

Time Complexity: O(logN) because it is using inbuilt sqrt function
Auxiliary Space: O(1)