Tetrahedral Numbers

A number is termed as a tetrahedral number if it can be represented as a pyramid with a triangular base and three sides, called a tetrahedron. The n^th tetrahedral number is the sum of the first n triangular numbers.
The first ten tetrahedral numbers are: 
1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ...
 

Formula for n^th tetrahedral number: 
 

Tn = (n * (n + 1) * (n + 2)) / 6

Proof: 
 

The proof uses the fact that the nth tetrahedral 
number is given by,

Trin = (n * (n + 1)) / 2

It proceeds by induction.

Base Case
T1 = 1 = 1 * 2 * 3 / 6

Inductive Step
Tn+1 = Tn + Trin+1

Tn+1 = [((n * (n + 1) * (n + 2)) / 6] + [((n + 1) * (n + 2)) / 2]

Tn+1 = (n * (n + 1) * (n + 2)) / 6

Below is the implementation of above idea : 
 

// CPP Program to find the
// nth tetrahedral number
#include <iostream>
using namespace std;

int tetrahedralNumber(int n)
{
    return (n * (n + 1) * (n + 2)) / 6;
}

// Driver Code
int main()
{
    int n = 5;
    
    cout << tetrahedralNumber(n) << endl;

    return 0;
}

// Java Program to find the
// nth tetrahedral number
class GFG {
    
// Function to find Tetrahedral Number
static int tetrahedralNumber(int n)
{
    return (n * (n + 1) * (n + 2)) / 6;
}

// Driver Code
public static void main(String[] args)
{
    int n = 5;
    
    System.out.println(tetrahedralNumber(n));
}
}

// This code is contributed by Manish Kumar Rai.

# Python3 Program to find the
# nth tetrahedral number

def tetrahedralNumber(n):
    
    return (n * (n + 1) * (n + 2)) / 6

# Driver Code
n = 5
print (tetrahedralNumber(n))

// C# Program to find the
// nth tetrahedral number
using System;

public class GFG{
    
    // Function to find Tetrahedral Number
    static int tetrahedralNumber(int n)
    {
        return (n * (n + 1) * (n + 2)) / 6;
    }
    
    // Driver code
    static public void Main ()
    {
        int n = 5;
    
        Console.WriteLine(tetrahedralNumber(n));
    }
}

// This code is contributed by Ajit.

<?php
// PHP Program to find the
// nth tetrahedral number

function tetrahedralNumber($n)
{
    return ($n * ($n + 1) * ($n + 2)) / 6;
}

// Driver Code
    $n = 5;

    echo tetrahedralNumber($n);
    
// This code is contributed by mits
?>

<script>

// JavaScript Program to find the
// nth tetrahedral number

// Function to find Tetrahedral Number
function tetrahedralNumber(n)
{
    return (n * (n + 1) * (n + 2)) / 6;
}
 
// Driver code
     let n = 5;
    document.write(tetrahedralNumber(n));
    
    // This code is contributed by code_hunt.
</script>

Output: 
 

35

Time Complexity: O(1).

Space complexity: O(1) since using constant variables