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:
    v=[0,1]
    for i in range(0,20):
        print v
        v += [0,0]
        v = [v[1]] + [(i-1)*v[i-1]+(i+1)*v[i+1] for i in range(1,len(v)-1)]


ETA: ANoI's explanation



Comments:

None: tan(x) actually
2006-03-14T21:41:26Z
If you want to evaluate the exact automaton in my post that should be v=[1,0,1]. That'll fix the extraneous 1+.. -- Dan, aka sigfpe
11011110:

11011110:
2006-03-15T01:07:00Z
I edited the post to change it from 1+tan(x) to tan(x) by starting with [0,1] instead of [1,1].