Diagonal/Non-diagonal Grid Filling

[Leroy Quet]

OK, here is a question and idea for a sequence related to the recently 
discussed problem.

With the rules of the puzzle (non-diagonal alternating with diagonal move 
to empty squares) on the (2n)-by-(2n) grid, it has been shown that the 
grid can be filled entirely (and we can start anywhere), if we allow the 
path to cross itself (at diagonal sections of the path).

So, for the (2n)-by-(2n) grid, what is the minimum number of crossings 
needed to be able to land on every square of the grid exactly once?

Sequence begins: 0, 1,...

4-by-4 example, with one crossing:

1  3  5  6
2  4  7  8
12 10 9 15
11 13 1416

Or we can ask about the sequence where a(n) relates to the n-by-n grid 
(instead of the (2n)-by-(2n) grid), but we want to know the minimum 
number of crossings to get the maximum possible number of landed-on 
squares.

thanks,
Leroy Quet