**Latest figures**

FG has posted <a href="http://dl.dropbox.com/u/217239/sol_rectangle.html">a solution for the 11x1 rectangle</a>, attaining 101/27 = 3.740740...

More results:  
7x3 rectangle is 56/15 = 3.733333...  
7x2 rectangle is 67/18 = 3.722222...  
8x1 rectangle is 26/7 = 3.714285...  
6x1 rectangle is 85/23 = 3.695652...  
7x1 rectangle is 11/3 = 3.666666...  
5x1 rectangle is 25/7 = 3.571428...

I have removed specific solutions from this answer, as they have been superseded by FG's results.