Abstract
Let , where is the sum of binary digits function. The sequence is the well-known Prouhet–Thue–Morse sequence. In this note we study the sequence , where and for we define recursively as follows: . We prove several results concerning arithmetic properties of the sequence . In particular, we prove non-vanishing of for , automaticity of the sequence for each m, and other results.
Acknowledgments
We are grateful to two anonymous referees for helpful suggestions regarding the content and presentation of this work. We are also grateful to Agnieszka Dutka for proofreading and LATE X support and to Piotr Miska for useful remarks.
Disclosure statement
No potential conflict of interest was reported by the author(s).