a new problem related to partitions, dissections and polyominoes

[Max]

On 2/22/06, Edwin Clark <eclark at math.usf.edu> wrote:

> A018808 Number of lines through at least 2 points of an n X n grid of
> points.
>
> 0, 0, 6, 20, 62, 140, 306, 536, 938, 1492, 2306, 3296, 4722, 6460,
> 8830, 11568, 14946, 18900, 23926, 29544, 36510, 44388, 53586, 63648,
> 75674, 88948, 104374, 121032, 139966, 160636, 184466, 209944, 239050,
> 270588, 305478, 342480, 383370, 427020

There is an interesting related sequence/problem.

Given an an n X n grid of points on a plane.
Let a(n) be the number of pairs of lines such that
1) lines in each pair are distinct;
2) each line passes through at least 2 points of the grid;
3) there are no points of the grid "in between" these lines.

Would anybody be interested in computing this sequence?

Max

P.S. Somewhat formal definition: there is no points "in between" two
lines if there exists a continuous motion (i.e., translation and
rotation) of one line into the other where no intermediate line passes
through a point of the grid.