[seqfan] Re: How many squares can you make from n points in the plane?
Kurz, Sascha
Sascha.Kurz at uni-bayreuth.de
Tue Oct 5 15:50:37 CEST 2021
> Exact values
> Values for n <= 9 are exact and are the same for a and b
I will send a proof for n <= 12 later on.
> Lower bounds
> In particular, a(37)>=117, a(48)>=198, a(49)>=207, a(50)>=216.
Maybe a(25)>=51, a(36)>=109, a(49)>=207 would be good
examples since they beat the square grid construction.
> Asymptotic behavior
For the lower bound using the square grid one might mention A002415
We have the following improved upper bound:
The maximal number of isosceles right triangles in a set of n points in the plane, see A186926,
was upper bounded by floor(2/3 *(n-1)^2-5/3) in [AFR]. Since each square contains 4 such triangles
we obtain the upper bound floor(2/3 *(n-1)^2-5/3)/4,which is roughly n^2/6.
Best,
Sascha
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