How many ways can we divide line segment of length n into length 2 and 3 segments

Question

Given a line segment of length $n$, determine the number of divisions of this line segment into black or white segments of length 3 and red, green or blue segments of length 2.
I tried using recurrence relation $a_n = 3a_{n-3} + 2a_{n-2}$ but I realised it's not correct, since for example, let $n = 5$, then I treat $3*blue + 2*blue$ and $2*blue+3*blue$ as two different decompositions, which is not correct.
Any ideas how to solve it?

Answer 1

You have the coefficients reversed as the length 3 segments come in 2 colors, and the length 2 segments come in 3 colors. Also the configuration you state can't actually occur because the length 3 segments and the length 2 segments have disjoint sets of colors, so $3*blue+2*blue$ is not possible, as the length 3 piece would have to be black or white.

So your relation ought to really be: $a_n=2a_{n-3}+3a_{n-2}$.