**Question Corner and Discussion Area**

The tetrahedronal numbers: 1, 4, 10, 20, 35, . . . are derived by adding the triangular numbers: 1 + 3 + 6 + 10 + 15 + · · · + (These numbers are all expressible as binomial coefficients. If we let denote then)(n+1)/2. At one time I derived a formula for the sum through themth 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,

This part of the site maintained by (No Current Maintainers)

Last updated: April 19, 1999

Original Web Site Creator / Mathematical Content Developer: Philip Spencer

Current Network Coordinator and Contact Person: Joel Chan - mathnet@math.toronto.edu

Go backward to Calculating Digits of Pi in Other Bases

Go up to Question Corner Index

Go forward to Existence of Shapes with Irrational Dimensions

Switch to text-only version (no graphics)

Access printed version in PostScript format (requires PostScript printer)

Go to University of Toronto Mathematics Network
Home Page