Jumping Farther: Grid Game

[Leroy Quet]

Regarding the game described in rec.puzzles and sci.math at:

http://groups.google.com/group/rec.puzzles/browse_thread/thread/9204248e8b1
87950/761ef98ba098ab8a?fwc=1

For an n-by-n grid, what is the maximum possible score?
(Or, what is the array of maximum scores where a(m,n) is for an m-by-n 
grid?)
How does this sequence begin? Is it in the EIS?
Is there a simple way to determine the maximum score?

thanks,
Leroy Quet