Brik's sequence: a strange recursion

We study the properties of the sequence of words $(B_i)$, where $B_1=101$ and $B_{i+1}=B_i{C}_i$ for $i\ge 1$, where $C_i$ is $B_i$ with the first $i$ symbols removed, and the infinite binary sequence $\mathbf{b}=10101101011011101\cdots$ of which all the $B_i$ are prefixes. We show that $\mathbf{b}$ 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.