The Hankel determinants of an automatic sequence need not be automatic

Allouche, Han and Niederreiter asked whether the sequence of Hankel determinants of a $d$-automatic sequence must be $d$-regular, or must become $d$-automatic after reduction modulo an integer. We answer the first question in the negative, and the second one for the reduction modulo $3$. Let $\alpha$ be the root of $f^4+f^2-Tf+1=0$ in $\mathbb{F}_3((1/T))$ with $|\alpha|<1$, the quartic of Mills and Robbins, and let $S$ be its coefficient sequence, which is $3$-automatic by Christol's theorem. The Hankel determinants of $S$ vanish in runs whose lengths are unbounded but of the order of the square root of their position, because the partial quotients of $\alpha$ are unbounded while its irrationality exponent is $2$; and a lemma of Schlage-Puchta forbids runs in that regime in an automatic sequence. Hence these Hankel determinants are $k$-automatic for no $k$, and the integer Hankel determinants of any lift of $S$ are $k$-regular for no $k$.