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Centered dodecahedral number

The centered dodecahedral number is a fascinating mathematical concept that arises in number theory. It is a sequence of numbers that can be represented as a 3-dimensional arrangement of spheres in the shape of a dodecahedron (a polyhedron with twelve pentagonal faces). In this article, we will explore the concept of centered dodecahedral numbers, understand the problem statement, provide a standard pseudocode and algorithm with explanations, demonstrate an example with resultant output, and finally, analyze the time complexity of the code.

Problem Statement

The problem is to generate the first 'k' centered dodecahedral numbers and display them. Each centered dodecahedral number is calculated using the formula: (2n + 1) * (5n^2 + 5n + 1), where 'n' represents the nth centered dodecahedral number.

Pseudocode

To generate the first 'k' centered dodecahedral numbers, we can follow the pseudocode below:

``````function centeredDodecahedralNo(k):
for n from 0 to k-1:
result = (2 * n + 1) * (5 * n^2 + 5 * n + 1)
display result
``````

Algorithm Explanation

1. Start the centeredDodecahedralNo function with parameter 'k', which represents the number of centered dodecahedral numbers to be generated.
2. Use a loop to iterate from 'n' = 0 to 'n' = k-1. Within the loop: a. Calculate the nth centered dodecahedral number using the formula provided: result = (2 * n + 1) * (5 * n^2 + 5 * n + 1). b. Display the calculated result.

Example: Let's consider 'k' = 10 to find the first ten centered dodecahedral numbers.

``centeredDodecahedralNo(10);``

Code Solution

Here given code implementation process.

``````// C Program for
// Centered dodecahedral number
#include <stdio.h>

void centeredDodecahedralNo(int k)
{
// Print all initial k centered dodecahedral number
for (int n = 0; n < k; ++n)
{
// Formula
//   (2n + 1) (5n² + 5n + 1)
// Calculate nth dodecahedral number
int result = (2 *n + 1) *(5 *(n *n) + 5 *n + 1);
// Display Calculated value
printf("  %d", result);
}
}
int main()
{
// Centered dodecahedral number are
// —————————————————————————————————————————————
//  1, 33, 155, 427, 909, 1661, 2743, 4215, 6137,
// 8569, 11571, 15203, 19525, 24597, 30479, 37231,
// 44913, 53585, 63307, 74139, 86141, 99373,
// 113895, 129767, 147049, 165801, 186083, 207955,
// 231477, 256709, 283711, 312543 ... etc
// n = 10
centeredDodecahedralNo(10);
return 0;
}``````

Output

``  1  33  155  427  909  1661  2743  4215  6137  8569``
``````// Java program for
// Centered dodecahedral number
public class CenteredDodecahedral
{
public void centeredDodecahedralNo(int k)
{
// Print all initial k centered dodecahedral number
for (int n = 0; n < k; ++n)
{
// Formula
//   (2n + 1) (5n² + 5n + 1)

// Calculate nth dodecahedral number
int result = (2 * n + 1) * (5 * (n * n) + 5 * n + 1);

// Display Calculated value
System.out.print(" " + result);
}
}
public static void main(String[] args)
{
// Centered dodecahedral number are
// —————————————————————————————————————————————
//  1, 33, 155, 427, 909, 1661, 2743, 4215, 6137,
// 8569, 11571, 15203, 19525, 24597, 30479, 37231,
// 44913, 53585, 63307, 74139, 86141, 99373,
// 113895, 129767, 147049, 165801, 186083, 207955,
// 231477, 256709, 283711, 312543 ... etc
// n = 10
}
}``````

Output

`` 1 33 155 427 909 1661 2743 4215 6137 8569``
``````// Include header file
#include <iostream>
using namespace std;
// C++ program for
// Centered dodecahedral number
class CenteredDodecahedral
{
public: void centeredDodecahedralNo(int k)
{
// Print all initial k centered dodecahedral number
for (int n = 0; n < k; ++n)
{
// Formula
//   (2n + 1) (5n² + 5n + 1)
// Calculate nth dodecahedral number
int result = (2 *n + 1) *(5 *(n *n) + 5 *n + 1);
// Display Calculated value
cout << " " << result;
}
}
};
int main()
{
// Centered dodecahedral number are
// —————————————————————————————————————————————
//  1, 33, 155, 427, 909, 1661, 2743, 4215, 6137,
// 8569, 11571, 15203, 19525, 24597, 30479, 37231,
// 44913, 53585, 63307, 74139, 86141, 99373,
// 113895, 129767, 147049, 165801, 186083, 207955,
// 231477, 256709, 283711, 312543 ... etc
// n = 10
return 0;
}``````

Output

`` 1 33 155 427 909 1661 2743 4215 6137 8569``
``````// Include namespace system
using System;
// Csharp program for
// Centered dodecahedral number
public class CenteredDodecahedral
{
public void centeredDodecahedralNo(int k)
{
// Print all initial k centered dodecahedral number
for (int n = 0; n < k; ++n)
{
// Formula
//   (2n + 1) (5n² + 5n + 1)
// Calculate nth dodecahedral number
int result = (2 * n + 1) * (5 * (n * n) + 5 * n + 1);
// Display Calculated value
Console.Write(" " + result);
}
}
public static void Main(String[] args)
{
// Centered dodecahedral number are
// —————————————————————————————————————————————
//  1, 33, 155, 427, 909, 1661, 2743, 4215, 6137,
// 8569, 11571, 15203, 19525, 24597, 30479, 37231,
// 44913, 53585, 63307, 74139, 86141, 99373,
// 113895, 129767, 147049, 165801, 186083, 207955,
// 231477, 256709, 283711, 312543 ... etc
// n = 10
}
}``````

Output

`` 1 33 155 427 909 1661 2743 4215 6137 8569``
``````package main
import "fmt"
// Go program for
// Centered dodecahedral number

func centeredDodecahedralNo(k int) {
// Print all initial k centered dodecahedral number
for n := 0 ; n < k ; n++ {
// Formula
//   (2n + 1) (5n² + 5n + 1)

// Calculate nth dodecahedral number
var result int = (2 * n + 1) * (5 * (n * n) + 5 * n + 1)

// Display Calculated value
fmt.Print(" ", result)
}
}
func main() {
// Centered dodecahedral number are
// —————————————————————————————————————————————
//  1, 33, 155, 427, 909, 1661, 2743, 4215, 6137,
// 8569, 11571, 15203, 19525, 24597, 30479, 37231,
// 44913, 53585, 63307, 74139, 86141, 99373,
// 113895, 129767, 147049, 165801, 186083, 207955,
// 231477, 256709, 283711, 312543 ... etc
// n = 10
centeredDodecahedralNo(10)
}``````

Output

`` 1 33 155 427 909 1661 2743 4215 6137 8569``
``````<?php
// Php program for
// Centered dodecahedral number
class CenteredDodecahedral
{
public	function centeredDodecahedralNo(\$k)
{
// Print all initial k centered dodecahedral number
for (\$n = 0; \$n < \$k; ++\$n)
{
// Formula
//   (2n + 1) (5n² + 5n + 1)

// Calculate nth dodecahedral number
\$result = (2 * \$n + 1) * (5 * (\$n * \$n) + 5 * \$n + 1);

// Display Calculated value
echo(" ".\$result);
}
}
}

function main()
{
// Centered dodecahedral number are
// —————————————————————————————————————————————
//  1, 33, 155, 427, 909, 1661, 2743, 4215, 6137,
// 8569, 11571, 15203, 19525, 24597, 30479, 37231,
// 44913, 53585, 63307, 74139, 86141, 99373,
// 113895, 129767, 147049, 165801, 186083, 207955,
// 231477, 256709, 283711, 312543 ... etc
// n = 10
}
main();``````

Output

`` 1 33 155 427 909 1661 2743 4215 6137 8569``
``````// Node JS program for
// Centered dodecahedral number
class CenteredDodecahedral
{
centeredDodecahedralNo(k)
{
// Print all initial k centered dodecahedral number
for (var n = 0; n < k; ++n)
{
// Formula
//   (2n + 1) (5n² + 5n + 1)

// Calculate nth dodecahedral number
var result = (2 * n + 1) * (5 * (n * n) + 5 * n + 1);

// Display Calculated value
process.stdout.write(" " + result);
}
}
}

function main()
{
// Centered dodecahedral number are
// —————————————————————————————————————————————
//  1, 33, 155, 427, 909, 1661, 2743, 4215, 6137,
// 8569, 11571, 15203, 19525, 24597, 30479, 37231,
// 44913, 53585, 63307, 74139, 86141, 99373,
// 113895, 129767, 147049, 165801, 186083, 207955,
// 231477, 256709, 283711, 312543 ... etc
// n = 10
}
main();``````

Output

`` 1 33 155 427 909 1661 2743 4215 6137 8569``
``````#  Python 3 program for
#  Centered dodecahedral number
class CenteredDodecahedral :
def centeredDodecahedralNo(self, k) :
n = 0
#  Print all initial k centered dodecahedral number
while (n < k) :
#  Formula
#    (2n + 1) (5n² + 5n + 1)
#  Calculate nth dodecahedral number
result = (2 * n + 1) * (5 * (n * n) + 5 * n + 1)
#  Display Calculated value
print(" ", result, end = "")
n += 1

def main() :
#  Centered dodecahedral number are
#  —————————————————————————————————————————————
#   1, 33, 155, 427, 909, 1661, 2743, 4215, 6137,
#  8569, 11571, 15203, 19525, 24597, 30479, 37231,
#  44913, 53585, 63307, 74139, 86141, 99373,
#  113895, 129767, 147049, 165801, 186083, 207955,
#  231477, 256709, 283711, 312543 ... etc
#  n = 10

if __name__ == "__main__": main()``````

Output

``  1  33  155  427  909  1661  2743  4215  6137  8569``
``````#  Ruby program for
#  Centered dodecahedral number
class CenteredDodecahedral
def centeredDodecahedralNo(k)
n = 0
#  Print all initial k centered dodecahedral number
while (n < k)
#  Formula
#    (2n + 1) (5n² + 5n + 1)
#  Calculate nth dodecahedral number
result = (2 * n + 1) * (5 * (n * n) + 5 * n + 1)
#  Display Calculated value
print(" ", result)
n += 1
end

end

end

def main()
#  Centered dodecahedral number are
#  —————————————————————————————————————————————
#   1, 33, 155, 427, 909, 1661, 2743, 4215, 6137,
#  8569, 11571, 15203, 19525, 24597, 30479, 37231,
#  44913, 53585, 63307, 74139, 86141, 99373,
#  113895, 129767, 147049, 165801, 186083, 207955,
#  231477, 256709, 283711, 312543 ... etc
#  n = 10
end

main()``````

Output

`` 1 33 155 427 909 1661 2743 4215 6137 8569``
``````// Scala program for
// Centered dodecahedral number
class CenteredDodecahedral()
{
def centeredDodecahedralNo(k: Int): Unit = {
var n: Int = 0;
// Print all initial k centered dodecahedral number
while (n < k)
{
// Formula
//   (2n + 1) (5n² + 5n + 1)

// Calculate nth dodecahedral number
var result: Int = (2 * n + 1) * (5 * (n * n) + 5 * n + 1);

// Display Calculated value
print(" " + result);
n += 1;
}
}
}
object Main
{
def main(args: Array[String]): Unit = {
var task: CenteredDodecahedral = new CenteredDodecahedral();
// Centered dodecahedral number are
// —————————————————————————————————————————————
//  1, 33, 155, 427, 909, 1661, 2743, 4215, 6137,
// 8569, 11571, 15203, 19525, 24597, 30479, 37231,
// 44913, 53585, 63307, 74139, 86141, 99373,
// 113895, 129767, 147049, 165801, 186083, 207955,
// 231477, 256709, 283711, 312543 ... etc
// n = 10
}
}``````

Output

`` 1 33 155 427 909 1661 2743 4215 6137 8569``
``````// Swift 4 program for
// Centered dodecahedral number
class CenteredDodecahedral
{
func centeredDodecahedralNo(_ k: Int)
{
var n: Int = 0;
// Print all initial k centered dodecahedral number
while (n < k)
{
// Formula
//   (2n + 1) (5n² + 5n + 1)

// Calculate nth dodecahedral number
let result: Int = (2 * n + 1) * (5 * (n * n) + 5 * n + 1);

// Display Calculated value
print(" ", result, terminator: "");
n += 1;
}
}
}
func main()
{
// Centered dodecahedral number are
// —————————————————————————————————————————————
//  1, 33, 155, 427, 909, 1661, 2743, 4215, 6137,
// 8569, 11571, 15203, 19525, 24597, 30479, 37231,
// 44913, 53585, 63307, 74139, 86141, 99373,
// 113895, 129767, 147049, 165801, 186083, 207955,
// 231477, 256709, 283711, 312543 ... etc
// n = 10
}
main();``````

Output

``  1  33  155  427  909  1661  2743  4215  6137  8569``
``````// Kotlin program for
// Centered dodecahedral number
class CenteredDodecahedral
{
fun centeredDodecahedralNo(k: Int): Unit
{
var n: Int = 0;
// Print all initial k centered dodecahedral number
while (n < k)
{
// Formula
//   (2n + 1) (5n² + 5n + 1)

// Calculate nth dodecahedral number
val result: Int = (2 * n + 1) * (5 * (n * n) + 5 * n + 1);

// Display Calculated value
print(" " + result);
n += 1;
}
}
}
fun main(args: Array < String > ): Unit
{
// Centered dodecahedral number are
// —————————————————————————————————————————————
//  1, 33, 155, 427, 909, 1661, 2743, 4215, 6137,
// 8569, 11571, 15203, 19525, 24597, 30479, 37231,
// 44913, 53585, 63307, 74139, 86141, 99373,
// 113895, 129767, 147049, 165801, 186083, 207955,
// 231477, 256709, 283711, 312543 ... etc
// n = 10
}``````

Output

`` 1 33 155 427 909 1661 2743 4215 6137 8569``

Resultant Output Explanation

The function will generate the first ten centered dodecahedral numbers: 1, 33, 155, 427, 909, 1661, 2743, 4215, 6137, and 8569.

Time Complexity

The time complexity of the provided code is O(k), where 'k' is the number of centered dodecahedral numbers to be generated. The loop runs 'k' times, and each iteration involves simple arithmetic operations to calculate the centered dodecahedral number, which takes constant time. As 'k' grows, the time taken by the loop increases linearly.

Finally

In this article, we explored the concept of centered dodecahedral numbers, understood the problem statement, provided pseudocode and an algorithm with explanations, demonstrated an example with output, and analyzed the time complexity of the code. Centered dodecahedral numbers are intriguing mathematical entities that have various applications in number theory and geometry. By implementing the provided code, one can generate and explore these unique numbers to gain insights into their properties and patterns.

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