Question Corner and Discussion Area

Tetrahedral and 4-Tetrahedral Numbers

The tetrahedronal numbers: 1, 4, 10, 20, 35, . . . are derived by adding the triangular numbers: 1 + 3 + 6 + 10 + 15 + · · · + (n)(n+1)/2. At one time I derived a formula for the sum through the mth term of the tetrahedronal numbers, and also the sum of the numbers for a 4-tetrahedron: 1, 5, 15, 35, 70, . . . , which are of course the sum of the tetrahedronal numbers. (Tetrahedronal numbers are best described by placing larger and larger triangular layers of 3-spheres under the previous triangular layer of spheres, forming a tetrahedronal pyramid.) Can you provide these formulas? Thanks.

You can prove this by induction noting that each equals 1, that the formula is correct when k=2, and the difference

You could also prove this by a more "brute-force" approach, using the formulas for the sums of powers:

and so on. To arrive at these formulas you can again proceed by brute force (make the intelligent guess that should be a polynomial of degree n+1 and plug in enough values to solve for the coefficients), or else employ a trick like the following:

Write plus terms involving lower powers of m. When you add up the LHS as m ranges from 0 to n, you get When you add up the RHS, you get plus terms involving down through This gives you

The terms cancel, and you are left with a formula that expresses in terms of down through .

For example, here is how to use the above technique to arrive at the formula for :

and simplifying gives the above formula. (Note that the reason n+1 appears at the end of the second line is that we are adding up the first line as m ranges from 0 to n; that means the final constant term "1" is added up n+1 times.)

Now that you have formulas for S_k(n), it is easy to find formulas for the tetrahedral numbers. For example,

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