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Tetrahedral Numbers
  • Last Updated : 05 Apr, 2021
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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 nth 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 nth 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 : 
 

C++




// 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




// 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.

Python




# 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#




// 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
// 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
?>

Javascript




<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).
 

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