Fiddling around with some recursion and the dragon curve and what should pop out but this exact sequence.
https://oeis.org/A072339
While it wasn't at all obvious to me that it would match a sequence like this, it is easy to demonstrate the connection. Take the odd numbers and use them as right turns and the even numbers as left turns.
Shame I don't have access to Knuth's work to see if the connection is described there.
@Unprovable
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