Call a pile of chips with no consecutive blues **good.^^ Let $b_n$ be the number of good piles of $n$ chips with a blue as the top chip. Let $c_n$ be the number of good piles of $n$ with blue not the top chip.
Note that $$b_{n+1}=c_n.\tag{1}$$ This is because we get a good arrangement of $n+1$ chips with blue on top by putting a blue on top of a good arrangement of height $n$ with blue not on top. Also, $$c_{n+1}=3(b_n+c_n),\tag{2}$$ since we get a good pile of $n+1$ with blue not on top by putting a chip of one of the $3$ non-blue colours on top of any good pile of $n$.
Bump the index of Recurrence (2) up by $1$. We get $$c_{n+2}=3b_{n+1} +3c_{n+1}.$$ Using Recurrence (1), we get $$c_{n+2}=3c_{n+1}+3c_n.\tag{3}$$
Solve Recurrence, (3) by some standard method. Or else if you are comfortable with matrix manipulation, solve the system of recurrences (1), (2).
Now that we have $c_n$, we know $b_n$. The number of good piles of $n$ chips is $b_n+c_n$.