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

[Ron Hardin]

I'll try a nXk table
T(n,k)=Number of nXk 0..1 arrays with exactly n+k-2 having value 1 and no three 1's forming an isoceles right triangle
which in the work of a few minutes looks interesting
..1...2....3....4....5....6....7....8....9..10.11.12
..2...6...10...21...34...62..100..171..276.458......
..3..10...24...24..107..236..499.1228.2753..........
..4..21...24...60..210..637.1840.5792...............
..5..34..107..210..768.1898.8211....................
..6..62..236..637.1898.7468.........................
..7.100..499.1840.8211..............................
..8.171.1228.5792...................................
..9.276.2753........................................
.10.458.............................................
(work not double-checked!)  I'll let it run further and see if any recurrences turn up in the columns (=rows) rhhardin at mindspring.com rhhardin at att.net (either)

 
      From: Neil Sloane <njasloane at gmail.com>
 To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu> 
 Sent: Sunday, April 24, 2016 4:30 PM
 Subject: [seqfan] No isosceles triangles in a square grid, continued
   
There is now an entry (A271914) for the array
T(m,n) = max number of points in mXn grid
such that no three form an isosceles triangle.

Initially I had conjectured that the answer was usually m+n-2,
(prompted by Giovanni's results for T(n,n))
but then Rob found many examples where it was m+n-1,
so then I thought it would always be either m+n-2 or m+n-1.
But then Rob pointed out that he had found T(6,9) =12 = m+n-3.

So now this is quite mysterious.

I still believe however that "If an mXn grid contains m+n points then
three of them form an isosceles triangle", since it sounds
so pretty. And I "almost" had a proof last night.

This is a new thread, there was too much baggage in the old one.

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