  We obtain the explicit evaluations of the Hankel determinants of the formal
power series $\prod_{k\geq 0}(1+Jx^{3^{k}})$ where $J={(\sqrt{-3}-1)}/2$, and
prove that the sequence of Hankel determinants is an aperiodic automatic
sequence taking value in $\{0, \pm 1, \pm J, \pm J^2\}$. This research is
essentially inspired by the works about Hankel determinants of Thue--Morse-like
sequences by Allouche, Peyri\`ere, Wen and Wen (1998), Bacher (2006) and the
first author (2013).
