Grid Game Question

[Leroy Quet]

This relates to the game posted at:

http://groups.google.com.sg/group/sci.math/browse_thread/thread/60eab6d87eab9d59#


As I posted:
What is the highest possible score for a given n-by-n 
grid? Is there an interesting sequence as n = 1,2,3,4, etc... Or is 
there an easy pattern to the highest possible scores? 

By the way, I am not seriously considering posting this sequence to the EIS, since I dislike sequences that are difficult to define and are defined elsewhere. (But that's just me.) Yet, I am curious about the values of the sequence anyway.

Thanks,
Leroy Quet