During the semester, I'm pretty sure I discovered a new sequence while playing around!

It satisfies the (hard to see in this picture) recurrence relation:

a₁ = 1
a_n = a_an-1 - 1

Building it is tricky, it has to be done simultaneously forwards and backwards. It's also not unique by any means, but if done right it has a proven explicit representation (below) which is about as nice as it gets. On top of that it can be extended to all real numbers!