Basic question with tiling tiles in a rectangle

Question

Find the number of ways of tiling a 2 by 10 rectangle with 1 by 2 tiles which can be placed in any orientation.

I’m just not sure how to approach it begin counting it

Answer 1

Since I already have a tiling program it's a 2 minute job to plug this in and get the answer '51'. If I didn't have a tiling program it would take maybe an hour to write a simple one.

Clarification: There are 89 ways to tile if you count rotations/reflections as distinct tilings. My program generates all 89 tilings, and for each it generates all distinct rotations and reflections, then assigns each a canonical score and discards tilings which aren't the smallest canonically.

Answer 2

I am not going to be providing a full solution but just a hint;

let $a_n$ denote number of ways to cover a $2*n$ rectangle with given tiles.

I'd suggest you to draw and see how $a_n$ = $a_{n-1}$ + $a_{n-2}$. (HINT; you can place the $2*1$ tiles in two ways, horizontally and vertically) and then find $a_0$ , $a_1$ and $a_2$ to get the rest or study some methods to solve recusrsion