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

Abstract: For any sequences $$\mathbf{u}={u(n)}<em>{n\geq0}, \mathbf{v}={v(n)}</em>{n\geq0},$$ we define $\mathbf{u}\mathbf{v}:={u(n)v(n)}<em>{n\geq0}$ and $\mathbf{u}+\mathbf{v}:={u(n)+v(n)}</em>{n\geq0}$. Let $f_i(x)~(0\leq i&lt; k)$ be sequence polynomials whose coefficients are integer sequences. We say an integer sequence $\mathbf{u}={u(n)}<em>{n\geq0}$ is a polynomial generated sequence if $${u(kn+i)}</em>{n\geq0}=f_i(\mathbf{u}),~(0\leq i&lt; k).$$ %Here we define $\mathbf{u}\mathbf{v}:={u(n)v(n)}<em>{n\geq0}$ and $\mathbf{u}+\mathbf{v}:={u(n)+v(n)}</em>{n\geq0}$ for any two sequences $$\mathbf{u}={u(n)}<em>{n\geq0}, \mathbf{v}={v(n)}</em>{n\geq0}.$$ In this paper, we study the polynomial generated sequences. Assume $k\geq2$ and $f_i(x)=\mathbf{a}<em>ix+\mathbf{b}_i~(0\leq i&lt; k)$. If $\mathbf{a}_i$ are $k$-automatic and $\mathbf{b}_i$ are $k$-regular for $0\leq i&lt; k$, then we prove that the corresponding polynomial generated sequences are $k$-regular. As a application, we prove that the Hankel determinant sequence $${\det(p</em>{i+j})<em>{i,j=0}^{n-1}}</em>{n\geq0}$$ 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.