(Almost) Filling Grid: Move By 1,2,1,2,...

[Leroy Quet]

In this thread (posted to both sci.math and rec.puzzles)

http://groups.google.com/group/rec.puzzles/browse_thread/thread/ba9d274b2ba
b9930/a66fd8406ef2615e#a66fd8406ef2615e

I ask about filling an n-by-n grid with positive integers (one integer at 
most per grid-square) such that 1 is in the upper-left square, each 
integer m is either below/ right of/ left of/ or above (m-1),
and each (2m-1) is 1 square from (2m), and each (2m) is 2 squares from 
(2m+1).

(Actually, my main topic in my post is moving by 3, by 2, by 3, by 2,..., 
instead of by 1, 2, 1, 2,.... But I think the 1,2,1,2,1,2,... situation 
is more interesting.)

So, two questions at least related to sequences come to mind regarding 
this topic.

1) Which n's are such that the n-by-n grid can be filled in entirely?
 For example, n=4 and n=5 work:
 1 15 16 14       1 10 20 11 21
 2  9  6 13       2  9 19 12 22
 4 10  5 11       5  6 17 14 24
 3  8  7 12       3  8 18 13 23
                  4  7 16 15 25

But 'triple_M_2' found that 34 is the highest possible for n=6.
(I find this claim totally believable, given that I repeatedly tried by 
hand to find a solution which completely filled the 6-by-6 grid, but many 
times only got as high as 34.)

2) What is the maximum possible number of squares that can be filled in 
for a n-by-n square?
For example, a(4)=16, a(5) = 25, a(6) = 34.

Also, we can ask about grids with toroidal topology, moving in other 
patterns (such as 2,3,2,3,...), starting with a move of 2 then 1 then 2 
then 1...(instead of 1 then 2 then 1 then 2...), non-square grids, 
allowing diagonal moves, etc.

thanks,
Leroy Quet