Determine the Unit Impulse Response of ARMA Filter for 8 First Indices
As you wrote correctly the function is given by:

$$ y \left[ n \right] = \frac{1}{2} y \left[ n - 1 \right] + \frac{1}{8} y \left[ n - 2 \right] + \frac{1}{2} x \left[ n - 2 \right] $$

Assuming initial conditions of 0 we can easily see that for unit impulse (Namely all zeros but $ x \left[ 0 \right] = 1 $:

$$ \begin{align*}
y \left[ 0 \right] &amp; = 0 \\
y \left[ 1 \right] &amp; = 0 \\
y \left[ 2 \right] &amp; = \frac{1}{2} \\
y \left[ 3 \right] &amp; = 0.5 * y \left[ 2 \right] = \frac{1}{4} \\
y \left[ 4 \right] &amp; = 0.5 * y \left[ 3 \right] + 0.125 * y \left[ 2 \right] = \frac{3}{16} \\
y \left[ 5 \right] &amp; = 0.5 * y \left[ 4 \right] + 0.125 * y \left[ 3 \right] = 
\frac{1}{8} \\
y \left[ 6 \right] &amp; = 0.5 * y \left[ 5 \right] + 0.125 * y \left[ 4 \right] = \frac{11}{128} \\
y \left[ 7 \right] &amp; = 0.5 * y \left[ 6 \right] + 0.125 * y \left[ 5 \right] = \frac{15}{256} \\
\end{align*}
$$

Since it has IIR Component it will go on infinitely.

MATLAB Simulation:



The full code can be found in my StackExchange Signal Processing Q43918 GitHub Repository.