I'm working on a homework assignment involving recursion and I'm having trouble finding an easy way to determine the initial conditions. Heres the problem:

We want to tile ann×1 strip with tiles of three types: 1×1 tiles that are dark-blue, light-blue,and red; 2×1 green tiles, and 3×1 sky-blue tiles. Now give a formula with initial conditions for the number of tilings, considering that blue tiles cannot be next to each other.

I can understand that the recurrence equation is:

$B\_n = B\_{n-1}+3B\_{n-2}+2B\_{n-3}+B\_{n-4}+B\_{n-5}$

And I've found initial conditions for

$B\_0=1$
$B\_1 = 3$
$B\_2 = 6$
$B\_3 = 17$

However, I found these but actually writing down all the possible combinations of tiles, but $B\_4$ is a huge possible list. Is there some method of combinatorics or permutations I can use to find the initial conditions for $B\_4$?