A Neighborhood of Infinity finds the power series expansion of tan(x) (somehow related to Joyce trees and Tremolo permutations) using a cute cellular-automaton-like iteration.

Here's the Python code I used to try it out:

If you want to evaluate the exact automaton in my post that should be v=[1,0,1]. That'll fix the extraneous 1+.
I edited the post to change it from 1+tan(x) to tan(x) by starting with [0,1] instead of [1,1].