Calculation of an impulse response of h[n]
That purple array is giving the impulse response over time. You can get it directly from the difference equation. Assume initial rest, $y[-1]=0$, then write out the impulse response for $n=0, 1, 2, ...$. If you do that you will get:
\begin{align}
 h[0]&amp;=1+0 \\
 h[1] &amp;= 0 + 0.5 \\
 h[2] &amp;= 0 + 0.25 \\
 h[3] &amp;= 0 + 0.125 \\
 ...
\end{align}