Octahedral number
An octahedral number is a figurate number that represents the number of spheres in an octahedron made by stacking layers of spheres. It can also be visualized as the sum of the first n triangular numbers. The formula for finding the nth octahedral number is n(2n² + 1) / 3.
Problem Statement
The task is to generate the first k octahedral numbers using the given formula and display the results.
Example
Let's take k = 5 as an example to understand the process of generating octahedral numbers.
- For n = 1: Octahedral number = (1 * (2 * (1 * 1) + 1)) / 3 = 1
- For n = 2: Octahedral number = (2 * (2 * (2 * 2) + 1)) / 3 = 6
- For n = 3: Octahedral number = (3 * (2 * (3 * 3) + 1)) / 3 = 19
- For n = 4: Octahedral number = (4 * (2 * (4 * 4) + 1)) / 3 = 44
- For n = 5: Octahedral number = (5 * (2 * (5 * 5) + 1)) / 3 = 85
Pseudocode
- Start with a function to calculate and display the first k octahedral numbers.
- Inside the function, use a loop to iterate from n = 1 to n = k.
- For each value of n, apply the octahedral number formula (n(2n² + 1) / 3) to find the result.
- Print the calculated result for each value of n.
Algorithm
FUNCTION octahedralNo(k):
FOR n = 1 to k:
result = (n * (2 * (n * n) + 1)) / 3
PRINT result
END FOR
END FUNCTION
FUNCTION main():
SET k = 10
CALL octahedralNo(k)
END FUNCTION
Code Solution
Here given code implementation process.
// C Program for
// Octahedral number
#include <stdio.h>
void octahedralNo(int k)
{
// Print all initial k octahedral number
for (int n = 1; n <= k; ++n)
{
// Formula
// n(2n²+1)
// —————————
// 3
// Calculate nth octahedral number
int result = (n *(2 *(n *n) + 1)) / 3;
// Display calculated result
printf(" %d", result);
}
}
int main()
{
// Octahedral number are
// —————————————————————————————————————————————
// 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,
// 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894,
// 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726,
// 13131, 14644, 16269, 18010, 19871, 21856, 23969,
// 26214, 28595, 31116, 33781, 36594, 39559 etc.
// k = 10
octahedralNo(10);
return 0;
}Output
1 6 19 44 85 146 231 344 489 670// Java program for
// Octahedral number
public class OctahedralNumber
{
public void octahedralNo(int k)
{
// Print all initial k octahedral number
for (int n = 1; n <= k; ++n)
{
// Formula
// n(2n²+1)
// —————————
// 3
// Calculate nth octahedral number
int result = (n * (2 * (n * n) + 1)) / 3;
// Display calculated result
System.out.print(" " + result);
}
}
public static void main(String[] args)
{
OctahedralNumber task = new OctahedralNumber();
// Octahedral number are
// —————————————————————————————————————————————
// 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,
// 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894,
// 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726,
// 13131, 14644, 16269, 18010, 19871, 21856, 23969,
// 26214, 28595, 31116, 33781, 36594, 39559 etc.
// k = 10
task.octahedralNo(10);
}
}Output
1 6 19 44 85 146 231 344 489 670// Include header file
#include <iostream>
using namespace std;
// C++ program for
// Octahedral number
class OctahedralNumber
{
public: void octahedralNo(int k)
{
// Print all initial k octahedral number
for (int n = 1; n <= k; ++n)
{
// Formula
// n(2n²+1)
// —————————
// 3
// Calculate nth octahedral number
int result = (n *(2 *(n *n) + 1)) / 3;
// Display calculated result
cout << " " << result;
}
}
};
int main()
{
OctahedralNumber *task = new OctahedralNumber();
// Octahedral number are
// —————————————————————————————————————————————
// 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,
// 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894,
// 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726,
// 13131, 14644, 16269, 18010, 19871, 21856, 23969,
// 26214, 28595, 31116, 33781, 36594, 39559 etc.
// k = 10
task->octahedralNo(10);
return 0;
}Output
1 6 19 44 85 146 231 344 489 670package main
import "fmt"
// Go program for
// Octahedral number
func octahedralNo(k int) {
// Print all initial k octahedral number
for n := 1 ; n <= k ; n++ {
// Formula
// n(2n²+1)
// —————————
// 3
// Calculate nth octahedral number
var result int = (n * (2 * (n * n) + 1)) / 3
// Display calculated result
fmt.Print(" ", result)
}
}
func main() {
// Octahedral number are
// —————————————————————————————————————————————
// 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,
// 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894,
// 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726,
// 13131, 14644, 16269, 18010, 19871, 21856, 23969,
// 26214, 28595, 31116, 33781, 36594, 39559 etc.
// k = 10
octahedralNo(10)
}Output
1 6 19 44 85 146 231 344 489 670// Include namespace system
using System;
// Csharp program for
// Octahedral number
public class OctahedralNumber
{
public void octahedralNo(int k)
{
// Print all initial k octahedral number
for (int n = 1; n <= k; ++n)
{
// Formula
// n(2n²+1)
// —————————
// 3
// Calculate nth octahedral number
int result = (n * (2 * (n * n) + 1)) / 3;
// Display calculated result
Console.Write(" " + result);
}
}
public static void Main(String[] args)
{
OctahedralNumber task = new OctahedralNumber();
// Octahedral number are
// —————————————————————————————————————————————
// 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,
// 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894,
// 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726,
// 13131, 14644, 16269, 18010, 19871, 21856, 23969,
// 26214, 28595, 31116, 33781, 36594, 39559 etc.
// k = 10
task.octahedralNo(10);
}
}Output
1 6 19 44 85 146 231 344 489 670<?php
// Php program for
// Octahedral number
class OctahedralNumber
{
public function octahedralNo($k)
{
// Print all initial k octahedral number
for ($n = 1; $n <= $k; ++$n)
{
// Formula
// n(2n²+1)
// —————————
// 3
// Calculate nth octahedral number
$result = (int)(($n * (2 * ($n * $n) + 1)) / 3);
// Display calculated result
echo(" ".$result);
}
}
}
function main()
{
$task = new OctahedralNumber();
// Octahedral number are
// —————————————————————————————————————————————
// 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,
// 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894,
// 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726,
// 13131, 14644, 16269, 18010, 19871, 21856, 23969,
// 26214, 28595, 31116, 33781, 36594, 39559 etc.
// k = 10
$task->octahedralNo(10);
}
main();Output
1 6 19 44 85 146 231 344 489 670// Node JS program for
// Octahedral number
class OctahedralNumber
{
octahedralNo(k)
{
// Print all initial k octahedral number
for (var n = 1; n <= k; ++n)
{
// Formula
// n(2n²+1)
// —————————
// 3
// Calculate nth octahedral number
var result = parseInt((n * (2 * (n * n) + 1)) / 3);
// Display calculated result
process.stdout.write(" " + result);
}
}
}
function main()
{
var task = new OctahedralNumber();
// Octahedral number are
// —————————————————————————————————————————————
// 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,
// 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894,
// 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726,
// 13131, 14644, 16269, 18010, 19871, 21856, 23969,
// 26214, 28595, 31116, 33781, 36594, 39559 etc.
// k = 10
task.octahedralNo(10);
}
main();Output
1 6 19 44 85 146 231 344 489 670# Python 3 program for
# Octahedral number
class OctahedralNumber :
def octahedralNo(self, k) :
n = 1
# Print all initial k octahedral number
while (n <= k) :
# Formula
# n(2n²+1)
# —————————
# 3
# Calculate nth octahedral number
result = int((n * (2 * (n * n) + 1)) / 3)
# Display calculated result
print(" ", result, end = "")
n += 1
def main() :
task = OctahedralNumber()
# Octahedral number are
# —————————————————————————————————————————————
# 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,
# 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894,
# 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726,
# 13131, 14644, 16269, 18010, 19871, 21856, 23969,
# 26214, 28595, 31116, 33781, 36594, 39559 etc.
# k = 10
task.octahedralNo(10)
if __name__ == "__main__": main()Output
1 6 19 44 85 146 231 344 489 670# Ruby program for
# Octahedral number
class OctahedralNumber
def octahedralNo(k)
n = 1
# Print all initial k octahedral number
while (n <= k)
# Formula
# n(2n²+1)
# —————————
# 3
# Calculate nth octahedral number
result = (n * (2 * (n * n) + 1)) / 3
# Display calculated result
print(" ", result)
n += 1
end
end
end
def main()
task = OctahedralNumber.new()
# Octahedral number are
# —————————————————————————————————————————————
# 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,
# 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894,
# 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726,
# 13131, 14644, 16269, 18010, 19871, 21856, 23969,
# 26214, 28595, 31116, 33781, 36594, 39559 etc.
# k = 10
task.octahedralNo(10)
end
main()Output
1 6 19 44 85 146 231 344 489 670// Scala program for
// Octahedral number
class OctahedralNumber()
{
def octahedralNo(k: Int): Unit = {
var n: Int = 1;
// Print all initial k octahedral number
while (n <= k)
{
// Formula
// n(2n²+1)
// —————————
// 3
// Calculate nth octahedral number
var result: Int = (n * (2 * (n * n) + 1)) / 3;
// Display calculated result
print(" " + result);
n += 1;
}
}
}
object Main
{
def main(args: Array[String]): Unit = {
var task: OctahedralNumber = new OctahedralNumber();
// Octahedral number are
// —————————————————————————————————————————————
// 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,
// 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894,
// 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726,
// 13131, 14644, 16269, 18010, 19871, 21856, 23969,
// 26214, 28595, 31116, 33781, 36594, 39559 etc.
// k = 10
task.octahedralNo(10);
}
}Output
1 6 19 44 85 146 231 344 489 670// Swift 4 program for
// Octahedral number
class OctahedralNumber
{
func octahedralNo(_ k: Int)
{
var n: Int = 1;
// Print all initial k octahedral number
while (n <= k)
{
// Formula
// n(2n²+1)
// —————————
// 3
// Calculate nth octahedral number
let result: Int = (n * (2 * (n * n) + 1)) / 3;
// Display calculated result
print(" ", result, terminator: "");
n += 1;
}
}
}
func main()
{
let task: OctahedralNumber = OctahedralNumber();
// Octahedral number are
// —————————————————————————————————————————————
// 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,
// 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894,
// 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726,
// 13131, 14644, 16269, 18010, 19871, 21856, 23969,
// 26214, 28595, 31116, 33781, 36594, 39559 etc.
// k = 10
task.octahedralNo(10);
}
main();Output
1 6 19 44 85 146 231 344 489 670// Kotlin program for
// Octahedral number
class OctahedralNumber
{
fun octahedralNo(k: Int): Unit
{
var n: Int = 1;
// Print all initial k octahedral number
while (n <= k)
{
// Formula
// n(2n²+1)
// —————————
// 3
// Calculate nth octahedral number
val result: Int = (n * (2 * (n * n) + 1)) / 3;
// Display calculated result
print(" " + result);
n += 1;
}
}
}
fun main(args: Array < String > ): Unit
{
val task: OctahedralNumber = OctahedralNumber();
// Octahedral number are
// —————————————————————————————————————————————
// 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,
// 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894,
// 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726,
// 13131, 14644, 16269, 18010, 19871, 21856, 23969,
// 26214, 28595, 31116, 33781, 36594, 39559 etc.
// k = 10
task.octahedralNo(10);
}Output
1 6 19 44 85 146 231 344 489 670Resultant Output Explanation
For k = 10, the first 10 octahedral numbers will be generated and displayed using the function octahedralNo(k). The output will be: 1 6 19 44 85 146 231 344 489 670
Time Complexity
The time complexity of this algorithm is O(k), where k is the number of octahedral numbers to be generated. Since we are using a loop to iterate from n = 1 to n = k and performing constant time operations within each iteration, the overall time complexity is directly proportional to k.
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