Hi! We've got an amazing crowd of interesting people here. We've got about 200 students registered, and I still have ten more applications to look at - they keep flooding in. I've been so busy registering people to the Azimuth Forum, saying hi, writing lectures and discussing the material that I'm a bit burnt out today.

A bunch of you have probably not had time to digest the material on Galois connections. The next lecture will be on Monday. I will spend more time explaining those, with more examples and applications.

But right now I'd like to pose a little puzzle, completely unrelated to the course material, to keep you entertained over the weekend.

Here's an interesting function:

$$ f(x) = \frac{1}{2\lfloor x \rfloor - x + 1} $$

\\(\lfloor x \rfloor\\) is the [floor function](https://en.wikipedia.org/wiki/Floor_and_ceiling_functions): it's the largest integer less than or equal to \\(x\\). For example,

$$ \lfloor 1.999 \rfloor = 1, \qquad \lfloor 2 \rfloor = 2 $$

We're starting to see the floor function in our discussion of Galois connections, and we'll see more of it, but that's not the point here. Here I want you to take the number \\(0\\) and keep hitting it with this function \\(f\\). I'll start you off:

$$ 0 $$

$$ f(0) = \frac{1}{2\lfloor 0 \rfloor - 0 + 1} = 1 $$

$$ f(1) = \frac{1}{2\lfloor 1 \rfloor - 1 + 1} = \frac{1}{2} $$

$$ f(1/2) = \frac{1}{2\lfloor \frac{1}{2} \rfloor - \frac{1}{2} + 1} = 2 $$

$$ f(2) = \frac{1}{2\lfloor 2 \rfloor - 2 + 1} = \frac{1}{3} $$

$$ f(1/3) = \frac{1}{2\lfloor \frac{1}{3} \rfloor - \frac{1}{3} + 1} = \frac{3}{2} $$

$$ f(3/2) = \frac{1}{2\lfloor \frac{3}{2} \rfloor - \frac{3}{2} + 1} = \frac{2}{3} $$

$$ f(2/3) = \frac{1}{2\lfloor \frac{2}{3} \rfloor - \frac{2}{3}+ 1} = 3 $$

So far we're getting this sequence:

$$ \frac{0}{1}, \frac{1}{2}, \frac{2}{1} , \frac{1}{3}, \frac{3}{2}, \frac{2}{3}, \frac{3}{1} , \dots $$

Here I'm writing all the numbers as fractions in [lowest terms](https://en.wikipedia.org/wiki/Irreducible_fraction), because this will help you see some patterns. And my puzzle is quite open-ended:

**Puzzle:** What are the interesting properties of this sequence? What patterns can you find?

If you already know from your studies of math, please _don't_ answer! Let others have the fun of discovering everything for themselves! One can discover a lot of amazing things by pondering this sequence.

Since a lot of you are programmers, the first step might be to compute the first hundred terms and show them to us! I urge you to show them in the way I just did, including writing numbers like \\(3\\) as \\(\frac{3}{1}\\).