Max values for coprimality game on grid

[Leroy Quet]

I am just curious.
As I say in the following sci.math post regarding my newest game:

http://groups.google.com/group/rec.puzzles/msg/72ac11da502679b2

I wonder what the maximum value is that is possible to get in 
the lower-right corner square for any n-by-n grid. 
(So, how does the sequence begin where the nth term is for the 
n-by-n grid?) 

Also I wonder about what the maximum number of odd integers 
can possibly occur in the entire grid, and the maximum 
number of even integers. (So, how does the sequence begin 
where the nth term is the maximum number of odds/evens for 
the n-by-n grid?) 

thanks,
Leroy Quet