Perfect linear complexity profile and Apwenian sequences

TL;DR

This paper reveals that perfect linear complexity profile sequences and Apwenian sequences are essentially the same, bridging two research communities and exploring implications for randomness and Hankel determinants.

Contribution

It establishes the equivalence between perfect linear complexity profile sequences and Apwenian sequences, unifying two previously separate areas of study.

Findings

Sequences are equivalent up to indexing.

Implications for randomness measures and Hankel determinants.

Bridging two research communities.

Abstract

Sequences with {\em perfect linear complexity profile} were defined more than thirty years ago in the study of measures of randomness for binary sequences. More recently {\em apwenian sequences}, first with values $\pm 1$, then with values in $\{0, 1\}$, were introduced in the study of Hankel determinants of automatic sequences. We explain that these two families of sequences are the same up to indexing, and give consequences and questions that this implies. We hope that this will help gathering two distinct communities of researchers.