[seqfan] Re: No isosceles triangles in a square grid, continued
Neil Sloane
njasloane at gmail.com
Wed Apr 27 13:07:42 CEST 2016
Ron, There are two separate problems being discussed:
No right isosceles triangles:
=====================
NORIT(m,n) = max no of 1's in mXn array so that no 3 form a right isosceles
triangle
- this array is not yet in OEIS (Because I don't know the initial values -
but I would like to)
A271906(n) = main diagonal of that array = max no of 1's in nXn array so
that no 3 form a right isosceles triangle. It is known that this grows like
O(n^1.99999) but not as fast as O(n^2) (!) (paper in preparation, with
Warren Smith, part of a general attack on problems of this class)
No isosceles triangles:
=================
A271914(m,n) = max no of 1's in mXn array so that no 3 form an isosceles
triangle, and
I conjecture A271907(m,n) <= m+n-1. The main diagonal is conjectured to be
= 2n-2 for n>1.
A271907(n) = main diagonal of that array = max no of 1's in nXn array so
that no 3 form an isosceles triangle,
which I conjecture to equal 2n-2 for n>1.
[The comment you quoted about "For (m,n) with 10 >= m >= n >= 2, the only
exceptions [to m+n-2] are..."
was referring to A271914(m,n). There are in fact infinitely
many exceptions, thanks to Rob's work. But (I conjecture) all values are <=
m+n-1.]
Best regards
Neil
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