Centered Cubic Number in C++

In this article, we will discuss Centered Cubic numbers in C++. Before discussing Centered Cubic numbers in C++. We must know about formulas, such as time complexity and space complexity.

Centered Cubic Number:

Centered cubic numbers, a unique kind of figurate numbers, represents the three-dimensional structure created by concentric cubic layers encircling a central point. These numbers indicate the total number of points that may be organized in a cube-centered arrangement with each additional layer growing outward symmetrically in all directions.

Simply put, a Centered Cube Number is the number of spheres (or lattice points) needed to create a centered cubic formation, in which point additions on the square sides of each layer beyond the center build upon the one before it. As a result, the 3D shape grows and its point distribution is symmetrical and visually balanced.

The Centered Cube numbers form a sequence where each term represents a specific pattern of increasing values. The first few terms in this sequence are 1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2355, 3149, 4131, 5331, and 6789…..

Formula:

Mathematical Formula for Centered Cube Numbers:

The n-th Centered Cube Number can be calculated using the formula

Where:

• Cn represents the n-th centered cubic number.
• The index of the centered cube number is denoted by n.
• At each layer, the cube's increasing dimensions are denoted by the term (2n+1).
• In a structured cubic formation, the complete distribution of points is taken into consideration by the expression (n^2+n+1).
• This formula efficiently determines how many points are needed to create a centered cube with n layers.

Example:

An Example of Computing.

Let us calculate the sixth centered cubic number to show how the formula works:

C6=(2(6)+1)×(6*6+6+1)

C6 =(12+1)×(36+6+1)

C6=13×43=341

341 is the 6th Centered Cube Number in the sequence.

Code:

Let us take an example to illustrate the Centered Cubic number in C++.

Output:

Enter the number of terms to compute: 10
Centered Cube Numbers for the first 10 terms:
C_1 = 9
C_2 = 35
C_3 = 91
C_4 = 189
C_5 = 341
C_6 = 559
C_7 = 855
C_8 = 1241
C_9 = 1729
C_10 = 2331   

Explanation:

A three-dimensional figurate number that forms concentric cubic layers around a central point is represented by the nth Centered Cube Number, which is computed using this C++ application. The nth term is calculated using the formula (2n+1)(n^2+n+1) by the function computeCenteredCubeNumber(int term). The user is prompted by the main() method to specify how many terms to compute. An error notice appears and the program ends if the input is not positive. Otherwise, the function is called for each term, and the results are printed, as the loop iterates from 1 to numberOfTerms. The application manages several calculations effectively, which ensures readability and ease of use.

Time Complexity:

The program has an O(n) time complexity, where n is the number of terms (numberOfTerms) that the user entered. The reason for this is that the program computes and prints the centered cube numbers by iterating through each term once in the for loop. Because the computeCenteredCubeNumber function runs in constant time O(1), the iteration, which runs n times, dominates the entire time complexity.

Space Complexity:

The program has O(1) space complexity, which means that regardless of the size of the input, it always utilizes the same amount of memory. Only a small number of variables, such as numberOfTerms and i, are stored by the program, and the computation of each term doesn't require additional space that expands with the input. As a result, no extra space that varies with the number of terms is utilized.

Conclusion:

In geometric visualization and three-dimensional number theory, centered cubic numbers are important. With their concentric layers, they resemble a cube and each adds a new point to the structure as a whole. The mathematical pattern of these numbers' growth can be effectively calculated with a formula or programmed into a computer for more involved computations.

Given their uses in combinatorial mathematics, crystallography, and 3D modeling, these numbers are an interesting topic for further exploration in number theory and computational geometry.