  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.
