Recursive combinatorial solution for number of sequences of {0,1,2} How do we approach a recursive solution to this combinatorial question?

In how many sequences of length $n$ that are made of the numbers ${0,1,2}$ that the sum of two consecutive digits is either 1, 2, or 3?

I have reached the fact that this question is equal to the question:

In how many sequences of length $n$ that are made of numbers ${0,1,2}$, where there are never two zero's next to each other or two 2's next to each other?

How does one solve this?

You can define 3 separate recursive sequences $n_{i,0}, n_{i,1}$ and $n_{i,2}$ that each count the number of sequences in $\{0,1,2 \}$ of length $i$ that end in 0, 1 or 2 respectively with no double 2s or double 0s. Clearly the total number of sequences of length $i$ satisfying your requirements is $n_{i} = n_{i,0}+n_{i,1}+n_{i,2}$. Now we have relations 
$n_{i+1, 0} = n_{i,1} + n_{i,2}$
$n_{i+1,1} = n_{i,0} + n_{i,1} + n_{i,2}$
$n_{i+1,2} = n_{i,0} + n_{i,1}$,
with initial data $n_{1,0} = n_{1,1} = _{1,2} = 1$.
Solve these recursively to get your general solution $n_{i}$.
