Diagram G and Recursive Sequences

From Geb

The author poses a problem of flipping diagram G as in a mirror, renumbering the nodes from left to right and finding an algebraic description of the new function. This is the third day I've spent thinking about it. Got it figured out? -- Nope! What troubles me most is that I can't think of any compelling reason why there couldn't be such a function. I spent about a week puzzling over it a couple years ago, shortly after finishing a BS in Mathematics and taking a year-long track in Abstract Algebra. I applied every trick I could think of to it, framing it in terms of recursive functions, digrams and number theory, trying to establish either a function or a proof that there couldn't exist such a function, all to no avail. Maybe some skilled programmer could try to brute-force a solution?