Finding the sum of all solutions $2x + 3y = n$ has exactly $2011$ non-negative integral solutions. Determine the SUM of the possible values of $n$.

The integer solutions to $2x+3y=1$ are given by
$$\left\{\begin{align*}
&x=-1+3k\\
&y=1-2k
\end{align*}\right.$$
for $k\in\Bbb Z$, so the integer solutions to $2x+3y=n$ are given by 
$$\left\{\begin{align*}
&x=-n+3k\\
&y=n-2k
\end{align*}\right.$$
for $k\in\Bbb Z$. Clearly $n>0$, so $x\ge 0$ iff $3k\ge n$ iff $k\ge\frac{n}3$, and $y\ge 0$ iff $2k\le n$ iff $k\le\frac{n}2$. Thus, for a given $n>0$ you get one solution to $2x+3y=n$ in non-negative integers for each integer $k$ satisfying $$\frac{n}3\le k\le\frac{n}2\;.\tag{1}$$
$(1)$ is equivalent to $$\left\lceil\frac{n}3\right\rceil\le k\le\left\lfloor\frac{n}2\right\rfloor\;,$$
so there are $\left\lfloor\frac{n}2\right\rfloor-\left\lceil\frac{n}3\right\rceil+1$ solutions. Thus, you’re looking for the sum of all positive integers $n$ such that
$$\left\lfloor\frac{n}2\right\rfloor-\left\lceil\frac{n}3\right\rceil+1=2011$$ or, equivalently,
$$f(n)\triangleq\left\lfloor\frac{n}2\right\rfloor-\left\lceil\frac{n}3\right\rceil=2010\;.$$
Clearly $$f(n)\approx\frac{n}2-\frac{n}3=\frac{n}6\;,$$ so the desired values of $n$ will be near $6\cdot2010=12060$. One of these values is $12060$, since $f(12060)=6030-4020=2010$. Now note that
$$\begin{align*}
f(k+6)&=\left\lfloor\frac{k+6}2\right\rfloor-\left\lceil\frac{k+6}3\right\rceil\\
&=\left(\left\lfloor\frac{k}2\right\rfloor+3\right)-\left(\left\lceil\frac{k}3\right\rceil+2\right)\\
&=\left\lfloor\frac{k}2\right\rfloor-\left\lceil\frac{k}3\right\rceil+1\\
&=f(k)+1
\end{align*}$$
for all $k$, so each arithmetic progression with constant difference $6$ contains exactly one term $n$ such that $f(n)=2010$, and there are therefore exactly $6$ such values of $n$. From here you should have no trouble finding the $6$ values of $n$ such that $f(n)=2010$ and their sum.
