The Hankel determinants of an automatic sequence need not be automatic
Source: hexagonmath.org/abs/2611.00032v1
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…
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