  Studying and comparing arithmetic properties of a given automatic sequence
and the sequence of coefficients of the composition inverse of the associated
formal power series (the formal inverse of that sequence) is an interesting
problem. This problem was studied before for the Thue-Morse sequence. In this
paper, we study arithmetic properties of the formal inverses of two sequences
closely related to the well-known Baum-Sweet sequence. We give the recurrence
relations for their formal inverses and we determine whether the sequences of
indices at which these formal inverses take value $0$ and $1$ are regular. We
also show an unexpected connection between one of the obtained sequences and
the formal inverse of the Thue-Morse sequence.
