On the regularity of the Hankel determinant sequence of the characteristic sequence of powers

For any sequences $$\mathbf{u}$={u(n)}{n\geq0}, $\mathbf{v}$={v(n)}{n\geq0},$wedefine$$\mathbf{u}$$\mathbf{v}$:={u(n)v(n)}{n\geq0}$and$$\mathbf{u}$+$\mathbf{v}$:={u(n)+v(n)}{n\geq0}$\mathrm{.}Let$f_i(x)~(0\leq i< k)$ be sequence polynomials whose coefficients are integer sequences. We say an integer sequence $\mathbf{u}=\{u(n){\}}_{n\ge 0}$ is a polynomial generated sequence if $$\{u(kn+i){\}}_{n\ge 0}=f_i(\mathbf{u}),(0\le i<k)\mathrm{.}$$ %Here we define $\mathbf{u}\mathbf{v}:=\{u(n)v(n){\}}_{n\ge 0}$ and $\mathbf{u}+\mathbf{v}:=\{u(n)+v(n){\}}_{n\ge 0}$ for any two sequences $\mathbf{u}=\{u(n){\}}_{n\ge 0},\mathbf{v}=\{v(n){\}}_{n\ge 0}\mathrm{.}$ In this paper, we study the polynomial generated sequences. Assume $k\ge 2$ and $f_i(x)={\mathbf{a}}_{i}x+{\mathbf{b}}_i(0\le i<k)$. If ${\mathbf{a}}_i$ are $k$-automatic and ${\mathbf{b}}_i$ are $k$-regular for $0\le i<k$, then we prove that the corresponding polynomial generated sequences are $k$-regular. As a application, we prove that the Hankel determinant sequence $\{\det (p_{i+j}{)}_{i,j=0}^{n-1}{\}}_{n\ge 0}$ is $2$-regular, where $\{p(n){\}}_{n\ge 0}=0110100010000\cdots$ is the characteristic sequence of powers 2. Moreover, we give a answer of Cigler’s conjecture about the Hankel determinants.