A Fibonacci type sequence with Prouhet–Thue–Morse coefficients

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.