  The Hankel determinants of a given power series $f$ can be evaluated by using
the Jacobi continued fraction expansion of $f$. However the existence of the
Jacobi continued fraction needs that all Hankel determinants of $f$ are
nonzero. We introduce {\it Hankel continued fraction}, whose existene and
unicity are guaranteed without any condition for the power series $f$. The
Hankel determinants can also be evaluated by using the Hankel continued
fraction.
  It is well known that the continued fraction expansion of a quadratic
irrational number is ultimately periodic. We prove a similar result for power
series. If a power series $f$ over a finite field satisfies a quadratic
functional equation, then the Hankel continued fraction is ultimately periodic.
As an application, we derive the Hankel determinants of several automatic
sequences, in particular, the regular paperfolding sequence. Thus we provide an
automatic proof of a result obtained by Guo, Wu and Wen, which was conjectured
by Coons-Vrbik.
