Computer assisted proof for Apwenian sequences related to Hankel determinants

TL;DR

This paper introduces a computer-assisted method to prove that certain sequences, including the Thue–Morse sequence, are Apwenian, meaning their Hankel determinants follow a specific oddness property for all positive integers.

Contribution

It develops an improved combinatorial approach combined with computer assistance to establish Apwenian properties of multiple sequences, extending previous results.

Findings

Confirmed the Apwenian property for several sequences

Enhanced combinatorial methods for Hankel determinant analysis

Validated results with computer-assisted proofs

Abstract

An infinite $\pm 1$-sequence is called {\it Apwenian} if its Hankel determinant of order $n$ divided by $2^{n-1}$ is an odd number for every positive integer $n$. In 1998, Allouche, Peyri\`ere, Wen and Wen discovered and proved that the Thue--Morse sequence is an Apwenian sequence by direct determinant manipulations. Recently, Bugeaud and Han re-proved the latter result by means of an appropriate combinatorial method. By significantly improving the combinatorial method, we prove that several other Apwenian sequences related to the Hankel determinants with Computer Assistance.