Note on the negative base xor sequence $n\oplus_{-b}(-n)$

TL;DR

This paper explores a relationship involving a negative base XOR-like sequence, providing proofs and generalizations using transducers, and addresses a conjecture related to sequence A178729.

Contribution

It introduces a novel proof and generalization of a conjecture about a negative base XOR sequence using transducers.

Findings

Established a relationship involving negative base XOR sequences.

Provided a proof and generalization of a conjecture in OEIS sequence A178729.

Utilized transducers to formalize and prove the sequence properties.

Abstract

We prove a relationship between certain integer expressions involving operators similar to the binary exclusive or. This gives a proof and generalization of a result conjectured about sequence A178729 in Sloane's Online Encyclopedia of Integer Sequences. We use transducers to prove this result.