[seqfan] Re: No right isosceles triangles in a square grid

[Giovanni Resta]

On 04/22/2016 12:28 PM, Neil Sloane wrote:

> Take a square grid of n X n points. What is
> the size of the largest subset S of these points
> such that no three of the points of S form a right isosceles triangle?
>
> That is, S must not contain 3 points A,B,C such that angle ABC
> = 90 degrees, and |AB| = |BC|.
> I am pretty sure that a(1)=1, a(2)=2, a(3)=4
> from
> XXX
> OOO
> OXO
> and I think a(4)=6.
>
> Wanted: enough terms to look this up in the OEIS
> and to add it if it is not there.

The first terms I got are 1, 2, 4, 6, 9, 11, 14, 17, 20, 23

A 8x8 matrix is

O X O O O O O X
X O O O O O O X
O O O X O O O X
O O X O O O O X
O O O O O O O X
O O O O O O O X
O O O O O O X O
X X X X X X O O

I also computed the similar sequence where all isosceles
triangles are forbidden.

The first terms I got are 1, 2, 4, 6, 8, 10, 12, 14, 14, 16, 20

A solution for 8x8:

O X X X X X O X
X O O O O O O X
X O O O O O O O
X O O O O O O O
X O O O O O O O
X O O O O O O O
O O O O O O O O
X X O O O O O O

It seems that these two sequences are not in OEIS.

Giovanni Resta