Automatic Sequences in Negative Bases and Proofs of Some Conjectures of Shevelev

We discuss the use of negative bases in automatic sequences. Recently the theorem-prover Walnut has been extended to allow the use of base (-k) to express variables, thus permitting quantification over Z instead of N. This enables us to prove results about two-sided (bi-infinite) automatic sequences. We first explain the theory behind negative bases in Walnut. Next, we use this new version of Walnut to give a very simple proof of a strengthened version of a theorem of Shevelev. We use our ideas to resolve two open problems of Shevelev from 2017. We also reprove a 2000 result of Shur involving bi-infinite binary words.