recurrence relation for the number of ways to make a pile of $n$ poker chips using red, white, and blue chips

Question

Find a recurrence relation for the number of ways to make a pile of $n$ poker chips using red, white, and blue chips and such that no two red chips are together (consecutive).

My workings:

Consider the color of the top chip, if it is red, then he one below cannot be red and the remaining $n-2$ chips give $a_{n-2}$ different ways.

If it is not red, then the remaining $n-1$ chips give $a_{n-1}$ different ways. Then I believe my recurrence relation is

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

My questions are the following:
1) are my workings correct?
2) should this be for n>2
3) is $a_1=3, a_2=8$ and why?

Answer 1

1) and 2) seem correct.

is $a_1=3, a_2=8$ and why?

• If you have two poker chips you have 8 ways to make a pile without two consecutive red chips: $wb,bw,rw,wr,rb,br,ww, bb$. Or you calculate all possible ways and substract the $rr$ combination: $a_2=3^2-1=8$
• Similar for $a_1$: The are 3 ways without two consecutive red chips: $r,b,w\Rightarrow a_1=3$