  Considering an arbitrary pair of distinct and non constant polynomials, $a$
and $b$ in $\mathbb{F}_2[t]$, we build a continued fraction in
$\mathbb{F}_2((1/t))$ whose partial quotients are only equal to $a$ or $b$. In
a previous work of the first author and Han (to appear in Acta Arithmetica),
the authors considered two cases where the sequence of partial quotients
represents in each case a famous and basic $2$-automatic sequence, both defined
in a similar way by morphisms. They could prove the algebraicity of the
corresponding continued fractions for several pairs $(a,b)$ in the first case
(the Prouhet-Thue-Morse sequence) and gave the proof for a particular pair for
the second case (the period-doubling sequence). Recently Bugeaud and Han
(arXiv:2203.02213) proved the algebraicity for an arbitrary pair in the first
case. Here we give a short proof for an arbitrary pair in the second case.
