Brik's sequence: a strange recursion

2026-05-08

AI Summary

This study investigates the structural properties and complexity of the infinite binary sequence $\mathbf{b}$ generated by the recursive rule $B_1 = 101$ and $B_{i+1} = B_i$ with its first $i$ symbols removed and appended. Combining tools from combinatorics on words, recursive sequence analysis, density estimation, and transcendental number theory, the authors establish for the first time that $\mathbf{b}$ is recurrent but not uniformly recurrent, exhibits exponential factor complexity, and is not morphic. Moreover, they prove that the density of 1s in $\mathbf{b}$ exists and is a transcendental number. These results uncover novel dynamical and combinatorial features of $\mathbf{b}$ and significantly advance the understanding of complexity in non-regular recursive sequences.

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.

Problem

• recurrent sequence
• factor complexity
• morphic sequence
• transcendental density
• binary sequence

Innovation

• recurrent sequence
• factor complexity
• non-morphic
• transcendental density
• binary word