Avoiding Colinear Points

[Leroy Quet]

At

http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&safe=off&threadm=cl89bq%
249su%241%40mercury.leidenuniv.nl&prev=

I describe a sequence and game based on the following idea.

If we have a rectangular grid of n lattice-points by n lattice-points, 
how many dots can we place at the lattice-points, at most one dot per 
lattice-point, so that no 3 or more dots are colinear (in any direction)?

I get the sequence:
1, 4, 6, 8, 10,..
And Richard Mathar has extended it:
12, 14, (15 or 16),...

The upper bound on each term is obviously 2n, since we can have at most 2 
dots in each row  and column.

A related sequence, which may or may not be in the EIS: How many ways are 
there (with any number of dots) to put dots in the n-by-n grid so that no 
3 or more dots are colinear?
(I leave the issue of counting or not counting the zero-dot case for the 
sequence's submitter.)

Another sequence:
How many ways are there to arrange the maximum number of dots in an 
n-by-n grid so that no dots are colinear?

A family of sequences:
For fixed m, how many ways are there to arrange m dots in an n-by-n grid 
so that no dots are colinear?

thanks,
Leroy Quet