Consider the following simple recursion:
\begin{align*} x_0 &= 0\\ x_{i+1} &= \left\lfloor \frac{3(x_i+1)}{2}\right \rfloor \end{align*}
and form a string $s$ from the least-significant bits of $x_i$, i.e., $s_i = x_i \bmod 2$.
The first few bits of $s$ are $0110001101...$
If I look at $s$, I notice that there are roughly as many $0$s and there are $1$s. In fact, I asked Mathematica to generate the first million digits of $s$, and plot a histogram of how often each of the 32 binary strings of length 5 appears within the first million digits of $s$ (code below):
and it certainly looks like each length-5 substring is equally likely to appear. There doesn't appear to be any bias when looking at other short substring lengths, either.
Is $s$ "random," in the sense that each substring of length $\ell$ has density $2^{-\ell}$ inside $s$? Is there general theory that explains this behavior?
Code for the experiment:
x0 = 0;
allbits =
Reap[For[i = 1, i < 1000000, i++, x0 = Floor[3*(x0 + 1)/2];
Sow[Mod[x0, 2]]];][[2]][[1]];
BarChart[Table[
SequenceCount[allbits, IntegerDigits[i, 2, 5],
Overlaps -> True], {i, 0, 31}], ImageSize -> Large]