  Sulanke and Xin developed a continued fraction method that applies to
evaluate Hankel determinants corresponding to quadratic generating functions.
We use their method to give short proofs of Cigler's Hankel determinant
conjectures, which were proved recently by Chang-Hu-Zhang using direct
determinant computation. We find that shifted periodic continued fractions
arise in our computation. We also discover and prove some new nice Hankel
determinants relating to lattice paths with step set $\{(1,1),(q,0),
(\ell-1,-1)\}$ for integer parameters $m,q,\ell$. Again shifted periodic
continued fractions appear.
