7. The number of non-negative integer solutions $(x, y, z)$ to the equation $x+2 y+3 z=2014$ is $\qquad$.
7. 339024.

Notice that, the number of non-negative integer solutions $(x, y)$ for the equation $x+2 y=k$ is $\left[\frac{k}{2}\right]+1$ (where $[x]$ denotes the greatest integer not exceeding the real number $x$).

Therefore, the number of non-negative integer solutions for the equation $x+2 y=2014-3 z$ is
$$
\begin{array}{l}
\sum_{z=0}^{671}\left(\left[\frac{2104-3 z}{2}\right]+1\right) \\
=\sum_{z=0}^{671}(1007-2 z)+\sum_{z=0}^{671}\left[\frac{z}{2}\right]+672 \\
=672 \times 1007-670 \times 672+335 \times 336 \\
=339024 .
\end{array}
$$