"Especially-Symmetric" Sequence

[Leroy Quet]

I sent this to sci.math (but mistyped the sequence's terms there):


Let an "especially-symmetric sequence" be a sequence (or string) of
terms (numbers, letters, etc.) where, if each a() is a term (specific
number, letter, etc):

s(1) = a(1);

s(m) is either {s(m-1),a(m),s(m-1)}
or {s(m-1),s(m-1)}

for all integers m >= 2.

(s(m) is a concatenation involving s(m-1).)

--
Examples:

If s(1) = a(1);
If we let s(2m) = {s(2m-1),a(m+1),s(2m-1)},
s(2m+1) = {s(2m),s(2m)}.

Then we have an infinite sequence for s(m) as m ->oo:

1,2,1,1,2,1,3,1,2,1,1,2,1,1,2,1,1,2,1,3,1,2,1,....

-
If we let s(m) = {s(m-1),(m-1),s(m-1)}, for all m>=1, then
the k_th term of any s(m), where k <= 2^m-1, is r(k),

where 2^r(k) is the highest power of 2 to divide k.

--

What are these sequences REALLY called? What obvious applications do
they have (such as when the terms of a continued fraction are of a
s(m))?

Thanks,
Leroy Quet