Hendecagonal Number In C++

Introduction

Numbers have intrigued mathematicians and developers for centuries. Among the many fascinating sequences is the hendecagonal numbers, which stand out for their geometric importance. These numbers depict an 11-gon or an 11-sided shape (a hendecagon) and are considered an extension of triangular, pentagonal, and hexagonal numbers.

In this tutorial, we will explore hendecagonal numbers, their significance, and potential applications in C++. Our goal is to offer straightforward and practical insights with minimal mathematical complexity to enhance understanding.

What are Hendecagonal Numbers?

Hendecagonal numbers are numerical representations of an 11-sided polygon, known as a hendecagon. These numbers exhibit a specific growth pattern as they progress through the series.

If you visualize an hendecagon (an 11-sided polygon) and position dots inside it following a specific pattern, you can generate a sequence of hendecagonal numbers. The first several hendecagonal numbers are as follows:

1, 11, 30, 58, 95, 141, 196, 260, 333, 415...

Each digit within this series adheres to a specific pattern in which additional points are incorporated into a growing hendecagon.

Properties of Hendecagonal Numbers:

Several properties of Hendecagonal Numbers in C++ are as follows:

• ✅ The series increases very fast, i.e., every number is much greater than the preceding number.
• ✅ They may be applied to tiling and geometric research.
• ✅ They have properties that can be applied to some number theory questions.
• ✅ Hendecagonal numbers are part of the family of polygonal numbers, a large family of numbers defined geometrically.

Hendecagonal Number Implementation in C++

We are going to develop a C++ application to display and generate hendecagonal numbers. The sequence will be generated using a simple iterative method.

#include <iostream>
using namespace std;

// Function to generate the nth hendecagonal number
int hendecagonalNumber(int n) {
    return (9 * n * n - 7 * n) / 2;
}

int main() {
    int terms;
    cout << "Enter the number of hendecagonal numbers to generate: ";
    cin >> terms;
    
    cout << "Hendecagonal Numbers: ";
    for (int i = 1; i <= terms; i++) {
        cout << hendecagonalNumber(i) << " ";
    }
    cout << endl;
    
    return 0;
}

Output:

Explanation:

Hendecagonal numbers are a sequence of polygonal numbers, a generalized set of numbers that are mathematically defined geometrically.

• ✅ Definition of Function : Function ‘hendecagonalNumber(int n)’ calculates the nth hendecagonal number using an expression.
• ✅ Input by User: The user, asking for the number of terms, takes the input.
• ✅ Loop Iteration: A loop execution is performed to iterate and print the hendecagonal numbers of the input number.
• ✅ Display: Successive numbers are printed.

Efficient Solution with Dynamic Programming

For large values of ‘n’, repetitive computation of the formula can be inefficient. To avoid this, we can employ dynamic programming in C++ to cache the computed results and avoid unnecessary recalculations.

We will proceed to develop a C++ application for the purpose of displaying and computing. The sequence will be determined through a simple iterative process.

#include <iostream>
#include <vector>
using namespace std;

void generateHendecagonalNumbers(int terms) {
    vector<int> hNumbers(terms);
    
    for (int i = 1; i <= terms; i++) {
        hNumbers[i - 1] = (9 * i * i - 7 * i) / 2;
    }
    
    cout << "Hendecagonal Numbers: ";
    for (int num : hNumbers) {
        cout << num << " ";
    }
    cout << endl;
}

int main() {
    int terms;
    cout << "Enter the number of hendecagonal numbers to generate: ";
    cin >> terms;
    generateHendecagonalNumbers(terms);
    return 0;
}

Output:

Why Use Dynamic Programming?

• ✅ It prevents repeated calculations through the storage of calculated values.
• ✅ It improves performance for the construction of large sequences.
• ✅ It uses arrays (or vectors ) to store results efficiently.

Applications of Hendecagonal Numbers:

Hendecagonal numbers, while less frequently spoken of than prime or Fibonacci numbers, possess special mathematical characteristics and uses:

• ✅ Geometry and Tessellations: It is used in the creation of patterns and tessellations.
• ✅ Mathematical Proof: It occurs in certain problems of number theory.
• ✅ Computer Graphics : It can be used to create polygon-based buildings.
• ✅ Simulations and Games: Employed to compute ordered placement in grid games.

Expanding the Concept:

If we would like to hear more, we can:

• ✅ Adapt the code to determine whether a given number is a hendecagonal number.
• ✅ Recursively generate hendecagonal numbers rather than iteratively.
• ✅ Graph the numbers to see the growth of the series.

Here is a basic function to check if a number is in the hendecagonal sequence:

#include <iostream>
#include <cmath>
using namespace std;

bool isHendecagonal(int num) {
    int n = (1 + sqrt(1 + 36 * num)) / 9;
    return ((9 * n * n - 7 * n) / 2) == num;
}

int main() {
    int num;
    cout << "Enter a number to check if it's a hendecagonal number: ";
    cin >> num;
    
    if (isHendecagonal(num))
        cout << num << " is a hendecagonal number." << endl;
    else
        cout << num << " is NOT a hendecagonal number." << endl;
    
    return 0;
}

Output:

Conclusion:

In conclusion, hendecagonal numbers are a very fascinating subject of study in number theory and computer programming. It is not such a widely used topic, but it is so fascinating that it is a very fascinating field to study. In this article, we:

• ✅ Conducted research on the subject of hendecagonal numbers.
• ✅ Prepared them using a C++ program.
• ✅ Improved the solution using dynamic programming.
• ✅ Researched actual uses of such numbers.