Yesterday, nobody contributed anything.  :-(
The official answer is: (2 * x + 2 ^ (2 * x)) / (1 - x).
fire solved the puzzle at 13:55:30.774; length: 24
turtle solved the puzzle at 15:19:34.749; length: 28
turtle solved the puzzle at 15:19:55.639; length: 23
turtle solved the puzzle at 15:20:16.247; length: 22
fire solved the puzzle at 16:29:14.712; length: 22