# Recurrence relations

• February 24th 2011, 10:45 AM
BrownianMan
Recurrence relations
$\textup{A 2 x 7 rectangle has tiling with 1 x 1 and 1 x 2 tiles (singletons and doubletons).}$
$\textup{How many such tilings of a 2 x 7 grid are there?}$

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

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

***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?