Brik's sequence: a strange recursion

TL;DR

The paper investigates the properties of a specific binary sequence generated through a recursive word sequence, revealing its recurrence, complexity, and transcendental density of ones.

Contribution

It introduces a new recursive sequence with unique combinatorial and number-theoretic properties, including non-morphic nature and transcendental density.

Findings

Sequence is recurrent but not uniformly recurrent.

Sequence has exponential factor complexity.

Density of 1's is transcendental.

Abstract

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.