Count of ways to write N as a sum of three numbers

Given a positive integer N, count number of ways to write N as a sum of three numbers. For numbers which are not expressible print -1.
Examples:
 

Input: N = 4 
Output: 3 
Explanation: 
( 1 + 1 + 2 ) = 4 
( 1 + 2 + 1 ) = 4 
( 2 + 1 + 1 ) = 4. 
So in total, there are 3 ways.
Input: N = 5 
Output: 6 
( 1 + 1 + 3 ) = 5 
( 1 + 3 + 1 ) = 5 
( 3 + 1 + 1 ) = 5 
( 1 + 2 + 2 ) = 5 
( 2 + 2 + 1 ) = 5 
( 2 + 1 + 2 ) = 5. 
So in total, there are 6 ways 
 

Approach: To solve the problem mentioned above if we take a closer look we will observe a pattern in solution to the question. For all the numbers that are greater than 2 we get a series 3, 6, 10, 15, 25 and so on, which is nothing but the sum of first N-1 natural numbers.
Below is the implementation of the above approach: 
 

// C++ program to count the total number of
// ways to write N as a sum of three numbers

#include <bits/stdc++.h>
using namespace std;

// Function to find the number of ways
void countWays(int n)
{
    // Check if number is less than 2
    if (n <= 2)
        cout << "-1";

    else {
        // Calculate the sum
        int ans = (n - 1) * (n - 2) / 2;

        cout << ans;
    }
}

// Driver code
int main()
{
    int N = 5;

    countWays(N);

    return 0;
}

// Java program to count the total number of
// ways to write N as a sum of three numbers
class GFG{

// Function to find the number of ways
static void countWays(int n)
{
    
    // Check if number is less than 2
    if (n <= 2)
    {
        System.out.print("-1");
    }
    else 
    {
        
        // Calculate the sum
        int ans = (n - 1) * (n - 2) / 2;
        System.out.print(ans);
    }
}

// Driver code
public static void main(String[] args)
{
    int N = 5;
    countWays(N);
}
}

// This code is contributed by Amit Katiyar

# Python3 program to count the total number of
# ways to write N as a sum of three numbers
def countWays(N): 

    # Check if number is less than 2
    if (N <= 2): 
        print("-1") 
    else:
        
        # Calculate the sum
        ans = (N - 1) * (N - 2) / 2
        
    print(ans) 

# Driver code 
if __name__ == '__main__': 
    
    N = 5
    countWays(N)

# This code is contributed by coder001

// C# program to count the total number of 
// ways to write N as a sum of three numbers 
using System;

class GFG{
    
// Function to find the number of ways 
static void countWays(int n) 
{ 
        
    // Check if number is less than 2 
    if (n <= 2) 
    { 
        Console.WriteLine("-1");
    } 
    else
    { 

        // Calculate the sum 
        int ans = (n - 1) * (n - 2) / 2; 
        Console.WriteLine(ans);
    } 
} 

// Driver code     
static void Main()
{
    int N = 5; 
    countWays(N); 
}
}

// This code is contributed by divyeshrabadiya07    

<script>

// Javascript program to count the total number of
// ways to write N as a sum of three numbers

// Function to find the number of ways
function countWays(n)
{
    // Check if number is less than 2
    if (n <= 2)
        document.write( "-1");

    else {
        // Calculate the sum
        var ans = (n - 1) * (n - 2) / 2;

        document.write( ans);
    }
}

// Driver code

var N = 5;
countWays(N);

</script>

6

Time Complexity: O(1)

Auxiliary Space: O(1) as constant space for variables is being used