The series ∞∑n=11n(n+1)=1 suggests it might be possible to tile a 1×1 square with nonrepeated rectangles of the form 1n×1n+1. Is there a known regular way to do this? Just playing and not having any specific algorithm, I got as far as the picture below, which serves more to get a feel for what I am looking for.
I think some theory about Egyptian fractions would help. It’s nice for instance in the center where 13+14+16+14=1. And on the right edge where 12+13+16=1.
Side note: The series is (11−12)+(12−13)+(13−14)+⋯. The similar looking (11−12)+(13−14)+(15−16)+⋯ sums to ln(2), and there is a nice picture for that, if you interpret ln(2) as an area under y=1x: