Minimal generating set size

[Edwin Clark]

On Thu, 9 Mar 2006, Dan Dima wrote:

>  Take all the points having integer coordinates (i,j,k) , 0 <= i,j,k <= N -
> inside a cube C of length N.
> 
> Determine a(N) as the size of a minimal generating set such that:
> There is a set A of size a(N) such that any point in C lies on some line
> generated by two points from A.
> Any other set of size smaller than a(N) does not have this property.
> I have seen some lower & upper bounds delivered for the 2D-case.
> Does anyone know if there is a general formula (for both 2D, 3D or even
> k-dim.space)?

Playing with this I found a somewhat related sequence which may be
described as follows:

Start with the "square" of lattice points S(n) = {(x,y), x=0..n,y=0..n}.  
Let G(n) be the set of all lattice points contained in lines determined by
distinct pairs of points in S(n). a(n) is the size of the largest square
with sides parallel to the axes and contained in G(n). That is, a(n) is 
the largest k such that some translate of S(k) is contained in G(n).

For n from 1 to 25 I obtain the sequence:

3, 6, 15, 22, 37, 46, 67, 92, 107, 138, 173, 192, 233, 278, 327, 354, 409, 
468, 531, 564, 633, 694, 783, 864, 907

The sequence is apparently not in the OEIS. Before I enter it I would
appreciate any idea for a better description of the sequence and a
confirmation that the terms are correct. Perhaps someone sees a formula
for a(n). 

--Edwin