Question

(Updated in response to comments below:)

I wrote a short Python program to count lines in small grids, which gave me enough data to search OEIS and find the answer at A018808A018809.

Here is the code; the only real trick is to do everything in projective coordinates rather than staying in the Cartesian world.

# Count lines through exactly two points in an n*n grid of points

# dictionary mapping lines to the number of times they occur
lines = []

{}

def meet((a,b,c),(d,e,f)):
    """The line ""Line through two points , or the point on two lines."""
    return ((b*f-c*e,c*d-a*f,a*e-b*d))

def same(l1,l2):
    """Do these two triples of coordinates represent the same line?"""
    return meet(l1,l2)==(0,0,0)

def add(l1):
    """Test whether a triple represents a new line, and if so add it to ""Update the list.""number of times line l1 has been generated."""
    if l1 == (0,0,0):
        return
    for l2 in lines:
        if same(l1,l2):
            lines[l2] += 1
            return
    lines.append(l1)

lines[l1] = 1

for n in range(1,7)range(1,8):
    lines = {}
    for a in range(n):
        for b in range(n):
            for c in range(n):
                for d in range(n):
                    add(meet((a,b,1),(c,d,1)))
    goodlines = 0
    for line,count in lines.items():
        if count == 2:
            goodlines += 1
    print len(lines)goodlines