Given the input and the output, how to determine the impulse response?
$\mathcal Z$-transforming the input and the output and then dividing them, we obtain the transfer function

$$\begin{array}{rl} H (z) &amp;= \dfrac{1 - z^{-1} - 4 z^{-2} + 4 z^{-3}}{1 - 3 z^{-1} + 2 z^{-2}}\\\\ &amp;= \dfrac{z^3 - z^2 - 4 z + 4}{z (z^2 - 3 z + 2)}\\\\ &amp;= \dfrac{(z-1)  (z-2) (z+2)}{z (z-1) (z-2)}\\\\ &amp;= \dfrac{z+2}{z} = \color{blue}{1 + 2 z^{-1}}\end{array}$$

Computing the inverse $\mathcal Z$-transform of $H (z)$, we finally obtain the impulse response.