In the coordinate plane, for a natural number $n$, let $a _ { n }$ be the number of points in the region
$$\left\{ ( x , y ) \mid 2 ^ { x } - n \leq y \leq \log _ { 2 } ( x + n ) \right\}$$
that satisfy the following conditions.\\
(a) The $x$-coordinate and $y$-coordinate are equal.\\
(b) The $x$-coordinate and $y$-coordinate are both integers.\\
For example, $a _ { 1 } = 2$ and $a _ { 2 } = 4$. Find the value of $\sum _ { n = 1 } ^ { 30 } a _ { n }$. [4 points]