Equations of the form $t(x+a)=t(x)$ and $t(x+a)=1-t(x)$ for Thue-Morse sequence

Abstract

For every $a\ge 1$ we give a recursion algorithm of building of set of solutions of equations of the form $t(x+a)=t(x)$ and $t(x+a)=1-t(x),$ where $\{t(n)\}$ is Thue-Morse sequence. We pose an open problem and two conjectures.

arXiv comment14 pages. Adding in proof of Theorem 2 in detail point 5

References in corpus (3)

• Several results on sequences which are similar to the positive integers
• On the Basis Polynomials in the Theory of Permutations with Prescribed Up-Down Structure
• On Connection between the Numbers of Permutations and Full Cycles with Some Restrictions on Positions and Up-Down Structure

Cited by in corpus (1)

• Pseudoperiodic Words and a Question of Shevelev