  In 1998, Allouche, Peyri\`{e}re, Wen and Wen showed that the Hankel
determinant $H_n$ of the Thue-Morse sequence over $\{-1,1\}$ satisfies
$H_n/2^{n-1}\equiv 1~(\mathrm{mod}~2)$ for all $n\geq 1$. Inspired by this
result, Fu and Han introduced \emph{apwenian} sequences over $\{-1,1\}$,
namely, $\pm 1$ sequences whose Hankel determinants satisfy $H_n/2^{n-1}\equiv
1~(\mathrm{mod}~2)$ for all $n\geq 1$, and proved with computer assistance that
a few sequences are apwenian. In this paper, we obtain an easy to check
criterion for apwenian sequences, which allows us to determine all apwenian
sequences that are fixed points of substitutions of constant length. Let $f(z)$
be the generating functions of such apwenian sequences. We show that for all
integer $b\ge 2$ with $f(1/b)\neq 0$, the real number $f(1/b)$ is
transcendental and its irrationality exponent is equal to $2$.
  Besides, we also derive a criterion for zero-one apwenian sequences whose
Hankel determinants satisfy $H_n\equiv 1~(\mathrm{mod}~2)$ for all $n\geq 1$.
We find that the only zero-one apwenian sequence, among all fixed points of
substitutions of constant length, is the period-doubling sequence. Various
examples of apwenian sequences given by substitutions with projection are also
given. Furthermore, we prove that all Sturmian sequences over $\{-1,1\}$ or
$\{0,1\}$ are not apwenian. And we conjecture that fixed points of substitution
of non-constant length over $\{-1,1\}$ or $\{0,1\}$ can not be apwenian.
