Conjectured properties of the Hankel determinants of the Prouhet–Thue–Morse sequence

Let $p,n\in\mathbb{N}$, and let

$$h_{p,n}=[c_{p+i+j-1}]_{1\leq i,j\leq n},$$

where ${\bf c}$ is the sequence considered in the paper. Define the Hankel determinant sequence by

$$H_{p,n}:=\operatorname{det}h_{p,n}.$$

Conjectured properties of the Hankel determinants. The following properties hold:

1. For each $p,n\in\mathbb{N}$, $H_{p,n}\in\{-1,0,1\}$.
2. We have

$$H_{0,n}=\begin{cases} 1, & n=\frac{1}{3}(4^{m}-1)\quad\text{for some }m\geq 2;\\ -1, & n\in\left\{\frac{2}{3}(4^{m}-1)+1,\frac{2}{3}(4^m-1)+1,\frac{4}{3}(4^m-1)+2\right\}\quad\text{for some }m\geq 1;\\ 0, & \text{otherwise}. \end{cases}$$

3. We have

$$H_{n,n}=\begin{cases} 1, & n=\sum_{i=1}^{m}4^{k_i},\quad\text{where }k_i<k_{i+1};\\ 0, & \text{otherwise}. \end{cases}$$

These assertions concern the arithmetic structure of Hankel determinants associated with the sequence ${\bf c}$; the paper presents them as based on numerical computations, so their general validity remains open.