Thue sequence

The Thue sequence, named after the Norwegian mathematician Axel Thue, is a recursively defined (recursive definition, a.k.a. inductive definition) binary string with some interesting properties.

We define the Thue sequence as the following. $T_0=0$, and for $n\ge 0$, $T_{n+1}=T_n+\overline{T_n}$ where $+$ represents concatenation and $\overline{T_n}$ is the complement of $T_n$ (all bits are flipped).

By this definition: $T_0=0$, $T_1=01$, $T_2=0110$, $T_3=01101001$, $T_4=0110100110010110$, and so on

Thue sequences have the following properties:

1. For every $n\ge 0$, the length of $T_n$ is $2^n$
2. For every $n\ge 1$, $T_n$ begins with $01$, and ends with $01$ (if $n$ is odd) or $10$ (if $n$ is even)
3. For every $n\ge 0$, $T_n$ never has more than two $0$s or two $1$s in a row.
   1. This builds upon the insight that two-in-a-row digits are first found at the site of concatenation ($T_2=0110$).
   2. Since all future $T_n$ are built upon prior Thue sequences and the prior sequences don’t have $000$ or $111$, the consecutive 3-digit sequence must occur at the junction (if exists). But the junction bits can only be $0110$ or $1010$, so there cannot be three or above consecutive $1$ or $0$ digits.