Update: I went ahead and added the method of obtaining the coefficients using DFT anyway. Like I mentioned above, this approach is quite tedious by hand, so I only work out the first few coefficients explicitly. In practice, these are almost always computed using numerical software. But for some people it may still be interesting to see a direct calculation rather than relying on shortcuts.

Here is the direct link to the new section on DFT: https://susam.net/fizz-buzz-with-cosines.html#dft

This is a joy.