fractal sequences

[Ralf Stephan]

> example: A066194 :: plot it as (x= k, y= a[k])
> (??)
> How many are at the same time a permutation over (all) the integers like
> A066194?

There is the evergreen A003188 (Gray code). It is 2-regular, which
I conjecture herewith A066194 to be, too. And I'll add, again, that
I suspect any self-similar sequence to have that property, as well as
having an ogf that is expressible as an infinite sum where k is in
the exponent of x. Most 2-regulars have an ogf with sum...x^2^k..., 
for example.

Can you prove it?


ralf