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Exercise V

Posted 06/01/2016
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This exercise is about the number of linear partitions of `2`-by-`n` grid points.

We consider a set of lattice points, `\{1,2,\cdots,n\}\times\{1,2\}`. For example, the set of `2`-by-`4` gird points are pictured as follows.

We want to partition these `2n` points into two nonempty parts by a straight line. An example of two different linear partitions of `2`-by-`4` grid points is pictured as follows.

Note that the area/size of grid points are not taken into consideration.

Suppose `a(n)` is the number of linear partitions of such `2\times n` grid points.

For example, `a(1)=1` and `a(2)=6`.

Find `a(n)`.