20. Let the positive integer $n \geqslant 2$, for the $2 \times n$ grid chain with $2 n$ nodes colored with red $(R)$, yellow $(Y)$, and blue $(B)$, three nodes at the left and right endpoints are already colored, as shown in the figure. If for the remaining $2 n-3$ nodes, each node is to be colored exactly one color, and adjacent nodes must be of different colors, find the number of different coloring methods.
For the $2 \times n$ grid chain with $2 n$ nodes colored in red $(R)$, yellow $(Y)$, and blue $(B)$, where the left endpoint is colored red or yellow, let the right endpoint be colored as $P, Q$.

Let $a_{n}$ be the number of colorings when $P=R$ (or $Y$), $Q=B$;
Let $b_{n}$ be the number of colorings when $P=B, Q=R$ (or $Y$);
Let $c_{n}$ be the number of colorings when $P=R, Q=Y$ or $P=Y, Q=R$.
Notice that (1) if there is no constraint on the right end, each additional cell has 3 different coloring methods, so $a_{n}+b_{n}+c_{n}=3^{n-1}$;
(2) by symmetry, flipping the figure upside down and swapping colors $R$ and $Y$, we have $a_{n}=b_{n}$;
(3) a simple recursion gives $a_{n}=2 b_{n-1}+c_{n-1}$.

Thus, $a_{n}=2 b_{n-1}+c_{n-1}=a_{n-1}+b_{n-1}+c_{n-1}=3^{n-2}$.
In summary, the number of different coloring methods that satisfy the conditions is $3^{n-2}$.