On the additive complexity of a Thue-Morse-like sequence

In this paper, we study the additive complexity ρt+(n) of a Thue-Morse-like sequence t=σ∞(0) with the morphism σ:0→01,1→12,2→20. We show that ρt+(n)=2⌊log2(n)⌋+3 for all integers n≥1. Consequently, (ρt+(n))n≥1 is a 2-regular sequence.