Is there a formula for the number of lines that contain exactly two points through an n x n
rectangular array of points?

$\begingroup$ see also my answer here: mathoverflow.net/q/240098/39495 $\endgroup$ – Moritz Firsching May 31 '16 at 11:59
(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 A018809.
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)): """Line through two points or point on two lines.""" return ((b*fc*e,c*da*f,a*eb*d)) def same(l1,l2): """Do these two triples represent the same line?""" return meet(l1,l2)==(0,0,0) def add(l1): """Update the 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[l1] = 1 for n in 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 goodlines

2$\begingroup$ This sequence is the number of lines through at least two points. I think the original question asked for the number of lines through exactly two points. $\endgroup$ – Will Orrick Oct 27 '09 at 16:25

$\begingroup$ Oops, you're right. Shouldn't be too hard to modify my program to detect and eliminate duplicate lines... $\endgroup$ – David Eppstein Oct 27 '09 at 16:29

2$\begingroup$ As a side note, you could have gotten this result by searching for (Exactly 2 points grid) in the OEIS, but not by searching for (Exactly two points grid). Annoying that the search works this way. $\endgroup$ – Kevin P. Costello Oct 27 '09 at 17:39
Maybe it would be easier to count the number of lines in nxn which pass through more than two points.
That's equivalent to asking how many 2ix2j rectangles there are in nxn, where i and j are relatively prime, except that it would overcount the lines that pass through 3 or more points.
Let R_k = the number of ki x kj rectangles in nxn, where i and j are relatively prime.
The number of lines passing through at least three points in nxn is R_2  R_3.
For $n \geq 3$ the gradient of these lines must be nonzero and finite. So it should be possible to give an answer expressed as a sum over all gradients which are reduced fractions $q/p$ with $n \leq q \leq n$, $\\,q \neq 0$ and $1 \leq p \leq n$.
For a given gradient $q/p$ you could count the number points $(x,y) \in \{1,2,\ldots,n\}^2$ that satisfy:
 $1 \leq x+q \leq n$,
 $y+p \leq n$,
 $xq \geq n+1$ or $xq \leq 0$ or $yq \leq 0$,
 $x+2q \geq n+1$ or $x+2q \leq 0$ or $y+2p \geq n+1$.
The first two dotpoints ensure that there is a second point $(x+q,y+p)$ on the line. The second two dotpoints ensure that there is not too many points on the line  that is, it ensures $(x+2q,y+2p)$ and $(xq,yp)$ are both not in $\{1,2,\ldots,n\}^2$.
To give it as a formula, the number of lines that contain exactly two points in $\{1,2,\ldots,n\}^2$ is \[\sum_{n \leq q \leq n} \sum_{1 \leq p \leq n} \chi(q,p) B_{q,p}\] for $n \geq 3$, where $\chi(q,p)=1$ if $\gcd(q,p)=1$, and $\chi(q,p)=0$ otherwise, and $B_{q,p}$ is the subset of $\{1,2,\ldots,n\}^2$ for which the above four dotpoints are satisfied.
This should have $O(n^4)$ time complexity (which is not great, but it's better than most formulae I typically deal with).