Shifted periodicity of Hankel determinants

Prove that the shifted periodicity of the Hankel determinant sequence H_n(M(x)^r) has order r when r is divisible by 3 and order 3r when r is congruent to 1 or 2 modulo 3.

Background

The paper applies Sulanke–Xin’s quadratic transformation to obtain shifted periodic continued fractions for generating functions associated with Motzkin convolution powers. The order of this shifted periodicity determines the recurrence structure used to evaluate the Hankel determinants.

Conjecture 8 asserts a residue-class-dependent general rule: order r for multiples of three and order 3r otherwise. The subsequent Theorem 9 reports verification of the conjectures for r<27, leaving the general statement beyond that computational range unresolved.