advantage of using impulse response function for LTI systems?
A sum of shifted and scaled impulse responses would look more like

$ y[n] = \sum_{k=-\infty}^n x[k] \cdot h[n-k] $

The impulse response of an accumulator is

$ h[n] = u[n] $

where $u[n]$ is the unit step function,

$u[n] = \begin{cases}
1 &amp; n \ge 0 \\
0 &amp; \mathrm{otherwise}
\end{cases}$

so for your example case of 

$ x[n] = \{1, 1, 2\} $

The result will be

$ \begin{align}
y[0] &amp;= \sum_{k=-\infty}^0 x[k] \cdot h[-k] \\
     &amp;= 1 \cdot 1 + 1 \cdot 0 + 2 \cdot 0 \\
     &amp;= 1 \\
y[2] &amp;= \sum_{k=-\infty}^2 x[k] \cdot h[2-k] \\
     &amp;= 1 \cdot 1 + 1 \cdot 1 + 2 \cdot 1 \\
     &amp;= 4
\end{align}$

and hopefuly you can see how $y[1]$ will be 2, and $y[n], n&gt; 2$ will be 4, as expected.

Your equations are incorrect for the sum of scaled and shifted impulse responses, and for your first input signal $x_0$.