Tiling a 2x7 Grid with 1x1 and 1x2 Tiles: Finding the Number of Tilings

[BrownianMan]

$$\textup{A 2 x 7 rectangle has tiling with 1 x 1 and 1 x 2 tiles (singletons and doubletons).}$$
$$\textup{How many such tilings of a 2 x 7 grid are there?}$$

$$\textup{Let }a_{n}\textup{ be the number of tilings of a 2 x n grid using 1 x 1 and 1 x 2 tiles so that the}$$
$$\textup{two rightmost squares are occupied by singletons, let }b_{n}\textup{ be the number of tilings}$$
$$\textup{ with one singleton and one doubleton in the rightmost squares, and let }c_{n}\textup{ be the}$$
$$\textup{ number of other tilings (two horizontal doubletons or one vertical doubleton). The}$$
$$\textup{problem asks for }a_{7}+b_{7}+c_{7}. \textup{ When one appends another column to the right side,}$$
$$\textup{forming a 2 x (n + 1) grid, either one can add 2 singletons or a vertical doubleton, or}$$
$$\textup{one can change any singletons in the nth column to doubletons. This yields:}$$

$$a_{n+1}=a_{n}+b_{n}+c_{n}$$
$$b_{n+1}=2a_{n}+b_{n}$$
$$c_{n+1}=2a_{n}+b_{n}+c_{n}$$

***doubletons may be placed vertically or horizontally.

Use the recurrence relations to obtain a numerical answer to the problem.

Not sure how do start this. Any help?

 

[vela]

Consider the case where n=1. Just count the different ways you can tile the grid to figure out a₁, b₁, and c₁.