Balanced maximum counting sequences

In this correspondence, we discuss a modified version of a method due to Bakos and, independently, to Robinson and Cohn for the construction of Gray sequences. We make use of this construction to prove the existence of so-called balanced cyclic half Gray sequences. Furthermore, we discuss a specific type of counting sequences, called maximum counting sequences, and we prove a conjecture of Robinson and Cohn with respect to the existence of balanced maximum counting sequences.