# Program to check if N is a Hexagonal Number or not

Given a number N, check if it is Hexagonal or not. If it is then print “Yes” otherwise output “No”.

Examples:

Input: N = 6
Output: Yes
Explanation:
6 is the second hexagonal number.

Input: N = 14
Output: No
Explanation:
The second hexagonal number is 6 while the third hexagonal number is 15. Hence 14 is not a hexagonal number.

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: To solve the problem mentioned above we have to note that the nth hexagonal Number is given by the formula: H(n) = n * (2n – 1). The formula indicates that the n-th hexagonal number depends quadratically on n. Therefore, find the positive integral root of N = H(n) equation.

Therefore, H(n) = nth hexagonal number and N is the given Number. Hence solve the following equation:

Here, H(n) = N
=> n * (2n – 1) = N
=> 2 * n * n – n – N = 0
The positive root of this equation is:
n = (1 + sqrt(8 N + 1)) / 4

After obtaining the value for n, check if it is an integer or not where n is an integer if n – floor(n) is 0.

Below is the implementation of the above approach:

## C++

 `// C++ Program to check ` `// if N is a Hexagonal Number ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Function to check ` `// if number is hexagonal ` `bool` `isHexagonal(``int` `N) ` `{ ` `    ``float` `val = 8 * N + 1; ` ` `  `    ``float` `x = 1 + ``sqrt``(val); ` ` `  `    ``// Calculate the value for n ` `    ``float` `n = (x) / 4; ` ` `  `    ``// Check if n - floor(n) ` `    ``// is equal to 0 ` `    ``if` `((n - (``int``)n) == 0) ` `        ``return` `true``; ` ` `  `    ``else` `        ``return` `false``; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `N = 14; ` ` `  `    ``if` `(isHexagonal(N) == ``true``) ` `        ``cout << ``"Yes"` `<< endl; ` ` `  `    ``else` `        ``cout << ``"No"` `<< endl; ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to check ` `// if N is a hexagonal number ` `import` `java.util.*; ` ` `  `class` `GFG { ` ` `  `// Function to check ` `// if number is hexagonal ` `static` `boolean` `isHexagonal(``int` `N) ` `{ ` `    ``float` `val = ``8` `* N + ``1``; ` ` `  `    ``float` `x = ``1` `+ (``float``)Math.sqrt(val); ` ` `  `    ``// Calculate the value for n ` `    ``float` `n = (x) / ``4``; ` ` `  `    ``// Check if n - floor(n) ` `    ``// is equal to 0 ` `    ``if` `((n - (``int``)n) == ``0``) ` `        ``return` `true``; ` ` `  `    ``else` `        ``return` `false``; ` `} ` ` `  `// Driver code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `N = ``14``; ` ` `  `    ``if` `(isHexagonal(N) == ``true``) ` `        ``System.out.println(``"Yes"``); ` `    ``else` `        ``System.out.println(``"No"``); ` `} ` `} ` ` `  `// This code is contributed by offbeat `

## Python3

 `# Python3 program to check ` `# if N is a hexagonal number ` `from` `math ``import` `sqrt ` ` `  `# Function to check ` `# if number is hexagonal ` `def` `isHexagonal(N): ` `     `  `    ``val ``=` `8` `*` `N ``+` `1` `    ``x ``=` `1` `+` `sqrt(val) ` ` `  `    ``# Calculate the value for n ` `    ``n ``=` `x ``/` `4` ` `  `    ``# Check if n - floor(n) ` `    ``# is equal to 0 ` `    ``if` `((n ``-` `int``(n)) ``=``=` `0``): ` `        ``return` `True` `    ``else``: ` `        ``return` `False` ` `  `# Driver code ` `if` `__name__ ``=``=` `'__main__'``: ` `     `  `    ``N ``=` `14` `     `  `    ``if` `(isHexagonal(N) ``=``=` `True``): ` `        ``print``(``"Yes"``) ` `    ``else``: ` `        ``print``(``"No"``) ` ` `  `# This code is contributed by BhupendraSingh `

## C#

 `// C# program to check if  ` `// N is a hexagonal number ` `using` `System; ` ` `  `class` `GFG{ ` ` `  `// Function to check ` `// if number is hexagonal ` `static` `bool` `isHexagonal(``int` `N) ` `{ ` `    ``float` `val = 8 * N + 1; ` `    ``float` `x = 1 + (``float``)Math.Sqrt(val); ` ` `  `    ``// Calculate the value for n ` `    ``float` `n = (x) / 4; ` ` `  `    ``// Check if n - floor(n) ` `    ``// is equal to 0 ` `    ``if` `((n - (``int``)n) == 0) ` `    ``{ ` `        ``return` `true``; ` `    ``} ` `    ``else` `    ``{ ` `        ``return` `false``; ` `    ``} ` `} ` ` `  `// Driver code ` `public` `static` `void` `Main(``string``[] args) ` `{ ` `    ``int` `N = 14; ` ` `  `    ``if` `(isHexagonal(N) == ``true``) ` `        ``Console.Write(``"Yes"``); ` `    ``else` `        ``Console.Write(``"No"``); ` `} ` `} ` ` `  `// This code is contributed by rutvik_56 `

Output:

```No
```

Time Complexity: O(1)

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Improved By : offbeat, bgangwar59, rutvik_56