General tetrahedral numbers

Start with a list of ones:

1, 1, 1, 1, 1, …

Taking the partial sums of this sequence gives consecutive numbers. That is, the nth number of the new series is the sum of the first n terms of the previous series.

1, 2, 3, 4, 5, …

If we take partial sums again, we get the triangular numbers.

1, 3, 6, 10, 15, …

They’re called triangle numbers because if we arrange coins in a triangle, with n on a side, the total number of coins is the nth triangular number.

If we repeat the process again, we get the tetrahedral numbers.

1, 4, 10, 20, 35, …

The nth tetrahedral number is the number of cannonballs is a tetrahedral stack, with each layer of the stack being a triangle.

We can produce the examples above in Python using the cumsum (cumulative sum) function on NumPy arrays.

    >>> import numpy as np
    >>> x = np.ones(5, int)
    >>> x
    array([1, 1, 1, 1, 1])
    >>> x.cumsum()
    array([1, 2, 3, 4, 5])
    >>> x.cumsum().cumsum()
    array([ 1, 3, 6, 10, 15])
    >>> x.cumsum().cumsum().cumsum()
    array([ 1, 4, 10, 20, 35])

We could continue this further, taking tetrahedral numbers of degree k. The sequences above could be called the tetrahedral numbers of degree k for k = 0, 1, 2, and 3.

Here’s Python code to find the nth tetrahedral number of a given dimension.

    def tetrahedral(n, dim):
        x = np.ones(n, int)
        for _ in range(dim):
             x = x.cumsum()
        return x[-1]

It turns out that

T(n, d) = \binom{n+d-1}{d}

and so we could rewrite the code above more simply as

    from math import comb

    def tetrahedral(n, dim):
        return comb(n + dim - 1, dim)

Related posts

Leave a Reply

Your email address will not be published.