5. For the Day of the Russian Flag, the seller decided to decorate the store window with 10 horizontal strips of fabric in three colors. At the same time, he follows two conditions:
1) strips of the same color should not hang next to each other;
2) each blue strip must hang between a white and a red one.

In how many ways can he do this?
5. Let's call the white and red strips of fabric green. Denote by $T_{n}$ the number of ways to decorate the shop window with $n$ strips of fabric. The first strip can only be a green strip. If the second strip is blue, then the following strips must be such that the first one is green. The total number of ways to hang the strips in this manner will be $T_{n-2}$. If the second strip is also green, then together with it, there will be $n-1$ remaining strips, which can be hung in $T_{n-1}$ ways. Thus, we get:

$$
T_{n}=T_{n-1}+T_{n-2}(*)
$$

Obviously, for $n=1$, $T_{1}=2$ (one strip can be either white or red), and for $n=2$, $T_{2}=2$ (two strips can be either white and red or red and white). Then all subsequent $T_{n}$ are found using the formula (*):

$$
\begin{gathered}
T_{3}=T_{2}+T_{1}=2+2=4, T_{4}=T_{3}+T_{2}=4+2=6, T_{5}=T_{4}+T_{3}=6+4=10, \\
T_{6}=T_{5}+T_{4}=10+6=16, T_{7}=T_{6}+T_{5}=16+10=26, T_{8}=T_{7}+T_{6}=26+16=42, \\
T_{9}=T_{8}+T_{7}=42+26=68, T_{10}=T_{9}+T_{8}=68+42=110 .
\end{gathered}
$$