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In 1998, Allouche, Peyri\`ere, Wen and Wen considered the Thue--Morsesequence, and proved that all the Hankel determinants of the period-doublingsequence are odd integral numbers. We speak of $t$-extension when the entriesalong the diagonal in the Hankel determinant are all multiplied by~$t$. Then weprove that the $t$-extension of each Hankel determinant of the period-doublingsequence is a polynomial in $t$, whose leading coefficient is the {\it onlyone} to be an odd integral number. Our proof makes use of the combinatorialset-up developed by Bugeaud and Han, which appears to be very suitable for thisstudy, as the parameter $t$ counts the number of fixed points of a permutation.Finally, we prove that all the $t$-extensions of the Hankel determinants of theregular paperfolding sequence are polynomials in $t$ of degree less than orequal to $3$.

On $t$-extensions of the Hankel determinants of certain automatic sequences