Create a recurrence relation for number of ways to construct something of length n

Question

Find a recurrence relation with initial condition for number of ways to create a frieze of length n can be formed.

The frieze is of width 1 and is being constructed with a supply of:

-red 1x1 tiles, green 1x2 tiles, brown 1x2 tiles, white 1x3 tiles

So far, by brute force method, I have found

$n_0 = 0, n_1 = 1, n_2 = 3, n_3 = 6, n_4 = 13, n_5 = 36$ (not sure if $n_5$ is correct)

But I could use help finding the recurrence relation. Any help would be appreciated

Answer 1

A frieze of length $n$ is:

$\cdot$ a frieze of length $n-1$ with a red tile attached

OR

$\cdot$ a frieze of length $n-2$ with a green OR brown tile attached

OR

$\cdot$ a frieze of length $n-3$ with a white tile attached.

Now all the ORs can be interpreted as $+$, with # of frieze of length $n$ being $a_n$. You get the recurrence:

$a_n=a_{n-1}+a_{n-2}+a_{n-2}+a_{n-3}=a_{n-1}+2a_{n-2}+a_{n-3}$.

You need the initial conditions $a_1,a_2,a_3$, which you have found already, to start using the recurrence. You can check that your $n_4$ satisfies this, but I get $n_5=28$