Two analogs of Thue-Morse sequence

TL;DR

This paper introduces two new sequences related to the Thue-Morse sequence, analyzing their structure, recurrence relations, and properties like cube-freeness, while highlighting open problems in the area.

Contribution

It presents two novel analogs of the Thue-Morse sequence based on binary and negabinary representations, including their recurrence formulas and structural properties.

Findings

The second sequence is proven to be cube-free.

The first sequence is shown to be quint-free.

Structural and recurrence formulas are established for both sequences.

Abstract

We introduce and study two analogs of one of the best known sequence in Mathematics : Thue-Morse sequence. The first analog is concerned with the parity of number of runs of 1's in the binary representation of nonnegative integers. The second one is connected with the parity of number of 1's in the representation of nonnegative integers in so-called negabinary (or in base $-2).$ We give for them some recurrent and structure formulas and prove that the second $(0,1)$-sequence is cube-free, while the first one is quint-free. Finally we consider several interesting unsolved problems.