Brik's sequence: a strange recursion
For researchers in combinatorics on words, this provides a new example with unusual properties, but the work is incremental.
The paper studies a recursively defined binary sequence and proves it is recurrent but not uniformly recurrent, has exponential factor complexity, is not morphic, and has a transcendental density of 1's.
We study the properties of the sequence of words $(B_i)$, where $B_1 = 101$ and $B_{i+1} = B_i C_i$ for $i \geq 1$, where $C_i$ is $B_i$ with the first $i$ symbols removed, and the infinite binary sequence ${\bf b} = 10101101011011101 \cdots$ of which all the $B_i$ are prefixes. We show that $\bf b$ is recurrent, but not uniformly recurrent; it has exponential factor complexity; it is not morphic; and the density of $1$'s exists and is transcendental.