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

For any sequences $\mathbf{u}=\{u(n)\}_{n\geq0}, \mathbf{v}=\{v(n)\}_{n\geq0},$ we define $\mathbf{u}\mathbf{v}:=\{u(n)v(n)\}_{n\geq0}$ and $\mathbf{u}+\mathbf{v}:=\{u(n)+v(n)\}_{n\geq0}$. Let $f_i(x)~(0\leq i< k)$ be sequence polynomials whose coefficients are integer sequences. We say an integer sequence $\mathbf{u}=\{u(n)\}_{n\geq0}$ is a polynomial generated sequence if $$\{u(kn+i)\}_{n\geq0}=f_i(\mathbf{u}),~(0\leq i< k).$$ %Here we define $\mathbf{u}\mathbf{v}:=\{u(n)v(n)\}_{n\geq0}$ and $\mathbf{u}+\mathbf{v}:=\{u(n)+v(n)\}_{n\geq0}$ for any two sequences $\mathbf{u}=\{u(n)\}_{n\geq0}, \mathbf{v}=\{v(n)\}_{n\geq0}.$ In this paper, we study the polynomial generated sequences. Assume $k\geq2$ and $f_i(x)=\mathbf{a}_ix+\mathbf{b}_i~(0\leq i< k)$. If $\mathbf{a}_i$ are $k$-automatic and $\mathbf{b}_i$ are $k$-regular for $0\leq i< k$, then we prove that the corresponding polynomial generated sequences are $k$-regular. As a application, we prove that the Hankel determinant sequence $\{\det(p_{i+j})_{i,j=0}^{n-1}\}_{n\geq0}$ is $2$-regular, where $\{p(n)\}_{n\geq0}=0110100010000\cdots$ is the characteristic sequence of powers 2. Moreover, we give a answer of Cigler's conjecture about the Hankel determinants.