We mentioned Young diagrams and tableaux in Exercise N. Here we color a Young diagram by rows using exactly three colors (red, blue and green) such that no adjacent rows have the same color. An exact `3`-colored Young diagram of shape `(5,3,3,2)` whose size is `13` and an exact `3`-colored of shape `(3,1,1,1)` whose size is `6` are drawn as examples in the following figure.
Clearly, a Young diagram could be exactly `3`-colored only if the Young diagram has at least three rows.
`a(n)` is the number of exact `3`-colored Young diagrams of size `n`. For example, `a(1)=0`, `a(2)=0`, `a(3)=6`.
The number of Young diagrams of size `n` is counted by the number of integer partitions of `n` which is Exercise A.
Find `a(n)`.