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 $fi(x)~(0\leq i< k)$ 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 $${u(kn+i)}{n\geq0}=fi(\mathbf{u}),~(0\leq i< k).$$ %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 $fi(x)=\mathbf{a}ix+\mathbf{b}i~(0\leq i< k)$. 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.