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 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).
Space complexity: O(1) since using constant variables
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