The game of Quod

[Dean Hickerson]

Joshua Zucker wrote:

> One sequence of relevance to this game is the number of squares in an
> nxn board with the four corners deleted.  He says that Denis Borris
> found that there are (n^4 - n^2 - 48n + 84)/12 of them, which is not
> in OEIS.

Mitch Harris replied:

> Call me kooky (or slow) but a quick check to me says that the 3x3 board
> with the 4 corners removed leaves 3^2 - 4 = 5 (unit) squares (and unit
> squares are the only ones that fit).

I think that we're supposed to start with an n by n array of points,
not squares.  Then delete the 4 corner points and count all squares
whose vertices are among the points; the sides of the squares don't
have to be horizontal or vertical.

So for n=3 we have 5 points (view in a fixed-width font):

     .
    ...
     .

The only square is formed by the 4 outer points, agreeing with a(3)=1.
For n=4 we have 12 points:

     ..
    ....
    ....
     ..

There are 5 unit squares, 4 tilted ones with sides sqrt(2), and 2 tilted
ones with sides sqrt(5), agreeing with a(4)=11.

Dean Hickerson
dean at math.ucdavis.edu