# Recurrence relations

• Feb 24th 2011, 09:45 AM
BrownianMan
Recurrence relations
\$\displaystyle \textup{A 2 x 7 rectangle has tiling with 1 x 1 and 1 x 2 tiles (singletons and doubletons).}\$
\$\displaystyle \textup{How many such tilings of a 2 x 7 grid are there?}\$

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

\$\displaystyle a_{n+1}=a_{n}+b_{n}+c_{n}\$
\$\displaystyle b_{n+1}=2a_{n}+b_{n}\$
\$\displaystyle 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?