Program to check if N is a Centered Pentadecagonal Number

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

Centered Pentadecagonal Number represents a dot in the centre and other dots surrounding it in successive Pentadecagonal(15-sided polygon) layers. The first few Centered pentadecagonal numbers are 1, 16, 46 …

Examples:

Input: N = 16 
Output: Yes 
Explanation:
Second Centered pentadecagonal number is 16.

Input: N = 20 
Output: No  

Approach:

1. The K^th term of the Centered pentadecagonal number is given as:

2. As we have to check that the given number can be expressed as a Centered Pentadecagonal 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 Pentadecagonal Number.

4. Else the number N is not a Centered Pentadecagonal 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 Pentadecagonal Number
bool isCenteredpentadecagonal(int N)
{
    float n
        = (16 + sqrt(120 * N + 105))
          / 30;

    // Condition to check if N is a
    // Centered Pentadecagonal Number
    return (n - (int)n) == 0;
}

// Driver Code
int main()
{
    // Given Number
    int N = 16;

    // Function call
    if (isCenteredpentadecagonal(N)) {
        cout << "Yes";
    }
    else {
        cout << "No";
    }
    return 0;
}

Java

// Java program for the above approach
class GFG{

// Function to check if number N is a
// Centered Pentadecagonal Number
static boolean isCenteredpentadecagonal(int N)
{
    float n = (float)(16 + Math.sqrt(120 * N +
                                     105)) / 30;

    // Condition to check if N is a
    // Centered Pentadecagonal Number
    return (n - (int)n) == 0;
}

// Driver Code
public static void main(String[] args)
{
    // Given Number
    int N = 16;

    // Function call
    if (isCenteredpentadecagonal(N))
    {
        System.out.println("Yes");
    }
    else
    {
        System.out.println("No");
    }
}
}

// This code is contributed by rutvik_56

Python3

# Python3 program for the above approach
import math

# Function to check if number N is a
# centered pentadecagonal number
def isCenteredpentadecagonal(N):

    n = (16 + math.sqrt(120 * N + 105)) / 30

    # Condition to check if N is a
    # centered pentadecagonal number
    return (n - int(n)) == 0

# Driver Code
N = 16

# Function call
if isCenteredpentadecagonal(N):
    print("Yes")
else :
    print("No")

# This code is contributed by ishayadav181

C#

// C# program for the above approach
using System;
class GFG{

// Function to check if number N is a
// centered pentadecagonal number
public static bool isCenteredpentadecagonal(int N)
{
    double n = (16 + Math.Sqrt(120 * N +
                               105)) / 30;

    // Condition to check if N is a
    // centered pentadecagonal number
    return (n - (int)n) == 0;
}

// Driver code
public static void Main()
{

    // Given number
    int N = 16;

    // Function call
    if (isCenteredpentadecagonal(N))
    {
        Console.WriteLine("Yes");
    }
    else
    {
        Console.WriteLine("No");
    }
}
}

// This code is contributed by divyeshrabadiya07

Javascript

<script>

// Javascript program for the above approach

// Function to check if number N is a
// Centered Pentadecagonal Number
function isCenteredpentadecagonal(N)
{
    var n = (16 + Math.sqrt(120 * N +
                          105)) / 30;

    // Condition to check if N is a
    // Centered Pentadecagonal Number
    return (n - (parseInt(n))) == 0;
}

// Driver Code

// Given Number
var N = 16;

// Function call
if (isCenteredpentadecagonal(N))
{
    document.write("Yes");
}
else
{
    document.write("No");
}

// This code is contributed by Kirti

</script>

Output:

No

Time Complexity: O(logN) since inbuilt sqrt function has been used

Auxiliary Space: O(1)