Program to check if N is a Star Number

Given an integer N, the task is to check if it is a star number or not.

Star number is a centered figurate number that represents a centered hexagram (six-pointed star) similar to Chinese checker game. The first few Star numbers are 1, 13, 37, 73 …

Examples:

Input: N = 13
Output: Yes
Explanation:
Second star number is 13.

Input: 14
Output: No
Explanation:
Second star number is 13, where as 37 is third.
Therefore, 14 is not a star number.



Approach:

  1. The Kth term of the star number is given as

    K^{th} Term = 6*K*(K-1) + 1

  2. As we have to check that the given number can be expressed as a star number or not. This can be checked as follows –

    => N = 6*K*(K-1) + 1
    => N = 6*K^{2} - 6*K + 1
    => K = \frac{6 + \sqrt{24*N + 12}}{12}

  3. Finally, check the value of computed using this formulae is an integer, which means that N is a star number.

Below is the implementation of the above approach:

C++

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// C++ implementation to check that
// a number is a star number or not
  
#include <bits/stdc++.h>
  
using namespace std;
  
// Function to check that the
// number is a star number
bool isStar(int N)
{
    float n
        = (6 + sqrt(24 * N + 12))
          / 6;
  
    // Condition to check if the
    // number is a star number
    return (n - (int)n) == 0;
}
  
// Driver Code
int main()
{
    int i = 13;
  
    // Function call
    if (isStar(i)) {
        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 star number or not
import java.io.*; 
import java.util.*; 
  
class GFG{ 
      
// Function to check that the
// number is a star number
static boolean isStar(int N)
{
    double n = (6 + Math.sqrt(24 * N + 12)) / 6;
  
    // Condition to check if the
    // number is a star number
    return (n - (int)n) == 0;
}
      
// Driver code
public static void main(String[] args) 
    int i = 13;
  
    // Function call
    if (isStar(i))
    {
        System.out.println("Yes");
    }
    else 
    {
        System.out.println("No");
    }
  
// This code is contributed by coder001

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Python3

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# Python3 implementation to check that 
# a number is a star number or not 
import math
  
# Function to check that the 
# number is a star number 
def isStar(N):
      
    n = (math.sqrt(24 * N + 12) + 6) / 6
      
    # Condition to check if the 
    # number is a star number
    return (n - int(n)) == 0
      
# Driver Code 
i = 13
  
# Function call
if isStar(i):
    print("Yes")
else:
    print("No")
  
# This code is contributed by ishayadav181

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C#

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// C# implementation to check that
// a number is a star number or not
using System;
  
class GFG{ 
      
// Function to check that the
// number is a star number
static bool isStar(int N)
{
    double n = (6 + Math.Sqrt(24 * N + 12)) / 6;
  
    // Condition to check if the
    // number is a star number
    return (n - (int)n) == 0;
}
      
// Driver code
public static void Main() 
    int i = 13;
  
    // Function call
    if (isStar(i))
    {
        Console.WriteLine("Yes");
    }
    else
    {
        Console.WriteLine("No");
    }
  
// This code is contributed by Code_Mech

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Output:

Yes

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