Last month, I was doodling in a notebook, and I wondered what the relationship was between the areas of quarter segments of progressively-larger circles as the radius increases by a unit. This first picture does a better job showing what I mean.
So the first circle has a radius of one, and the second circle has a radius of two, and I'm comparing a quarter of the first circle to the quarter of the difference between the first and second circles. Anyway, it turns out that you can solve for the area of any such segment with radius n using this sequence:
I thought that was interesting.