Centered Tetrahedral Number

What do you understand by a centered tetrahedral number? Let’s explore it in this article.

Firstly, what is a tetrahedral number?

A tetrahedral number is a figurate number that represents the number of spheres in a tetrahedron. It is also known as a triangular pyramid number. A tetrahedron is a three−dimensional geometric shape that has four triangular faces, six edges, and four vertices.

To find the nth tetrahedral number, you can use the formula:

Tn = (n * (n + 1) * (n + 2)) / 6

For example, the first few tetrahedral numbers are: 1,4,10,20,35,56,84

Look at the diagram below to have a clear understanding of tetrahedral numbers. Here we have A pyramid with a sides length of 5 containing 35 spheres. Each layer represents one of the first five triangular numbers.

Now, what is a centered tetrahedral number?

A centered tetrahedral number is a type of figurate number that represents the number of spheres needed to create a centered tetrahedron of a given size. A centered tetrahedron is a three−dimensional geometric shape that has a tetrahedron inside an octahedron, with one sphere at the center of each face.

The formula to find the nth−centered tetrahedral number is:

CTn = ((2n+1)(n^2+n+3))/3

Some of the initial centered tetrahedral numbers are 1, 5, 15, 35, 69, ….

Approach

Now, let’s convert the logic discussed above into a stepwise approach we will use in our code implementation.

• Specify the value of n, this can also be taken as user input.

• Use the formula CTn = ((2n+1)(n^2+n+3))/3 to calculate the nth−centered tetrahedral number.

• Print the calculation to the console.

C++ Code Implementation

Too much theory? Let’s get straight to code. Here is the c++ code implementation to calculate the nth−centered tetrahedral number.

Example

#include <iostream>
using namespace std;

int centeredTetrahedralNumber(int n) {
    return ((2*n+1)*(n*n+n+3))/3;
}

int main() {
    int n=9;
    cout << "The " << n << "th centered tetrahedral number is: " << centeredTetrahedralNumber(n) << endl;
    return 0;
}

Output

The 9th centered tetrahedral number is: 589

Time Complexity: O(1)

Space Complexity: O(1)

Conclusion

In this article, we have covered what is a tetrahedral number, also what is centered tetrahedral number. In addition, we also covered the logic to calculate an nth−centered tetrahedral number and its c++ code implementation. Hope you found the article helpful.