$T_2$ and $T_5$ are illustrated as follows. One could figure what $T_n$ looks like according to the examples.
$T_2$
$T_5$
A linear partition results from partitioning given points by a straight line into two nonempty parts. Note that the area/size of grid points are not taken into consideration.
An example of two different linear partitions of $T_5$ is pictured as follows.
Suppose $a(n)$ is the number of distinct linear partitions of $T_n$.
For example, $a(2)=3$.
Find $a(n)$.