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Centered Octahedral number

  • Last Updated : 19 Jul, 2021

We are given a number n, we need to find n-th centered octahedral number.
Description: A centered octahedral number is a figurate number. It counts the number of points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin. The same numbers are special cases of the Delannoy numbers, which count certain two-dimensional lattice paths.
The First Few Centered octahedral numbers (where n = 0, 1, 2, 3…….) are : 
1, 7, 25, 63, 129, 231, 377, 575, 833, 1159…………………………. 
Mathematics formula for nth Centered octahedral number: 
 

\begin{math}Co_{n} = \frac{(2n+1)(2n^2+2n+3)}{3}

Examples : 
 

Input : n = 6
Output : 377

Input : n = 15
Output : 4991

 

 

C++




// C++ Program to find nth
// Centered octahedral number
#include <bits/stdc++.h>
using namespace std;
 
// Function to find
// Centered octahedral number
 
int centeredOctahedral(int n)
{
    // Formula to calculate nth
    // Centered octahedral number
    // and return it into main function.
    return (2 * n + 1) * (2 * n * n + 2 * n + 3) / 3;
}
 
// Driver Code
int main()
{
    int n = 3;
    cout << centeredOctahedral(n) << endl;
 
    n = 9;
    cout << centeredOctahedral(n) << endl;
 
    return 0;
}

Java




// Java Program to find nth
// Centered octahedral number
import java.io.*;
 
class GFG
{
    // Function to find
    // Centered octahedral number
    static int centeredOctahedral(int n)
    {
         
    // Formula to calculate nth
    // Centered octahedral number
    // and return it into main function.
     
    return (2 * n + 1) *
           (2 * n * n + 2 * n + 3) / 3;
    }
 
    // Driver Code
    public static void main (String[] args)
    {
    int n = 3;
    System.out.print( centeredOctahedral(n));
    System.out.println();
    n = 9;
    System.out.print(centeredOctahedral(n));
    }
}
 
// This code is contributed by aj_36

Python3




# Python 3 Program to find nth
# Centered octahedral number
 
# Centered octahedral
# number function
def centeredOctahedral(n) :
     
    # Formula to calculate nth
    # Centered octahedral number
    # return it into main function.
    return (2 * n + 1) * (
            2 * n * n +
            2 * n + 3) // 3
 
# Driver Code
if __name__ == '__main__' :
         
    n = 3
    print(centeredOctahedral(n))
    n = 9
    print(centeredOctahedral(n))
 
# This code is contributed ajit

C#




// C# Program to find nth
// Centered octahedral number
using System;
 
public class GFG {
     
    // Function to find
    // Centered octahedral number
    static int centeredOctahedral(int n)
    {
         
        // Formula to calculate nth
        // Centered octahedral number
        // and return it into main function.
         
        return (2 * n + 1) *
            (2 * n * n + 2 * n + 3) / 3;
    }
 
    // Driver Code
    static public void Main ()
    {
        int n = 3;
        Console.WriteLine(
               centeredOctahedral(n));
 
        n = 9;
        Console.WriteLine(
              centeredOctahedral(n));
    }
}
 
// This code is contributed by m_kit.

PHP




<?php
// PHP Program to find nth
// Centered octahedral number
 
// Function to find
// Centered octahedral number
function centeredOctahedral($n)
{
    // Formula to calculate
    // nth Centered octahedral
    // number and return it
    // into main function.
    return (2 * $n + 1) *
           (2 * $n * $n +
            2 * $n + 3) / 3;
}
 
// Driver Code
$n = 3;
echo centeredOctahedral($n), "\n";
 
$n = 9;
echo centeredOctahedral($n), "\n";
 
// This code is contributed ajit
?>

Javascript




<script>
 
// Javascript Program to find nth
// Centered octahedral number
 
// Function to find
// Centered octahedral number
function centeredOctahedral(n)
{
     
    // Formula to calculate nth
    // Centered octahedral number
    // and return it into main function.
     
    return (2 * n + 1) *
           (2 * n * n + 2 * n + 3) / 3;
}
 
 
// Driver Code
 
var n = 3;
document.write(centeredOctahedral(n));
document.write("<br>");
 
n = 9;
document.write(centeredOctahedral(n));
 
// This code is contributed by Kirti
     
</script>
Output : 



63
1159

 

Time Complexity: O(1)
Auxiliary Space: O(1)

Reference: 
https://en.wikipedia.org/wiki/Centered_octahedral_number
 

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