Hankel continued fractions and Hankel determinants for $q$-deformed metallic numbers

Fix $n$ a positive integer. Take the $n$-th metallic number $\phi_n=\frac{n+\sqrt{n^2+4}}{2}$ (e.g. $\phi_1$ is the golden number) and let $\Phi_n(q)$ be its $q$-deformation in the sense of S. Morier-Genoud and V. Ovsienko. This is an algebraic continued fraction which admits an expansion into a Taylor series around $q=0$, with integral coefficients. By using the notion of Hankel continued fraction introduced by the first author in 2016 we determine explicitly the first $n+2$ sequences of shifted Hankel determinants of $\Phi_n$ and show that they satisfy the following properties: 1) They are periodic and consist of $-1,0,1$ only. 2) They satisfy a three-term Gale-Robinson recurrence, i.e. they form discrete integrable dynamical systems. 3) They are all completely determined by the first sequence. This article thus validates a conjecture formulated by V. Ovsienko and the second author in a recent paper and establishes new connections between $q$-deformations of real numbers and sequences of Catalan or Motzkin numbers.