Base 3/2 and Greedily Partitioned Sequences

TL;DR

This paper explores the relationship between base 3/2 representations and greedy partitions into 3-free sequences, revealing a fractal structure that connects these concepts and characterizes the Stanley cross-sequence.

Contribution

It uncovers a fractal structure in base 3/2 representations and proves their connection to the Stanley cross-sequence derived from greedy 3-free partitions.

Findings

Identifies a fractal structure in base 3/2 digit strings.

Shows that even integers in base 3/2 correspond to the Stanley cross-sequence.

Establishes a link between base 3/2 representations and greedy 3-free sequence partitions.

Abstract

We delve into the connection between base $\frac{3}{2}$ and the greedy partition of non-negative integers into 3-free sequences. Specifically, we find a fractal structure on strings written with digits 0, 1, and 2. We use this structure to prove that the even non-negative integers written in base $\frac{3}{2}$ and then interpreted in base 3 form the Stanley cross-sequence, where the Stanley cross-sequence comprises the first terms of the infinitely many sequences that are formed by the greedy partition of non-negative integers into 3-free sequences.