Palindromic Binary Sequence (again?)

[Leroy Quet]

I may have posted the below to seq.fan almost a year ago already, but I 
cannot recall.
In any case, the sequence of 1's and  0's is not in the EIS, so perhaps I 
did not post this before. (yet I did not use Superseeker.)
(I apologize if this is a repost here.)


Consider the sequence of 1's and 0's which is "super-palindromic", and 
has an interesting property if converted to a binary-represented real.

Let each A be a finite sequence of 0's and 1's, 
where:

A(1) = 1;

for k >= 2,

A(k) = A(k-1) A(k-1) , if a(k-1) = 0;

A(k) = A(k-1) 0 A(k-1) , if a(k-1) = 1;


where the sequences A(k) are concatenations involving A(k-1),

and a(k) is the k_th term of the sequence

limit{k->oo}  A(k).


The sequence begins:

1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,
1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,...


(Interesting in that the sequence is not only palindromic, but HOW it is 
palindromic is also palindromic {in a recursive manner}.) 

 
Now, consider the real x, where

x = sum{k=1 to oo} a(k)/2^k,

the binary-represented x having the k_th digit right of the decimal-point 
(the binary-point?) equal to a(k).

ie.
x = .101101010110101011010101101... 

Then the ratio of number of 0's in this total sequence ("total sequence" 
= the limit of A(k) as k approaches infinity) to the number of 1's in 
this total sequence is...
x.

---

There is at least one other real, y, with the ratio-property: its binary 
representaion has a ratio of 
y = (number of 0's in binary representation) 
/(number of 1's in binary representation).

Even though it is not palindromic, the y-sequence can be generated in a 
similar manner to the x-sequence.

B(1) = 1;

for k >= 2,

B(k) = B(k-1) B(k-1)   , if b(k-1) = 0;

B(k) = B(k-1) B(k-1) 0 , if b(k-1) = 1;

b(k) = k_th element of {limit k-> oo} B(k).

(I think this is right...)(??)

What is known about all reals with the "ratio property"?


thanks,
Leroy Quet