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%I #63 Dec 01 2016 01:33:04
%S 1,2,2,3,2,3,4,4,4,4,5,4,4,4,5,6,5,5,5,5,6,7,6,6,6,6,6,7,8,7,8,7,7,8,
%T 7,8,9,8,8,8,8,8,8,8,9,10,9,10,9,9,9,9,10,9,10
%N Symmetric array read by antidiagonals: T(n,k) (n>=1, k>=1) = maximal number of points that can be chosen in an n X k rectangular grid such that no three distinct points form an isosceles triangle.
%C It is conjectured that T(n,k) <= n+k-1.
%C The array is symmetric: T(n,k) = T(k,n).
%C The main diagonal T(n,n) appears to equal 2n-2 for n>1. (This diagonal is presently A271907, but if it really is 2n-2 that entry may be recycled.)
%C The triangle must have nonzero area (three collinear points don't count as a triangle).
%H Rob Pratt, <a href="/A271914/a271914.txt">Complete list of examples where T(n,k) != n+k-2 for 10 >= n >= k >= 2</a>. Note T(9,6) = T(6,9) = 12, which is n+k-3.
%F From _Chai Wah Wu_, Nov 30 2016: (Start)
%F T(n,k) >= max(n,k).
%F T(n,max(k,m)) <= T(n,k+m) <= T(n,k) + T(n,m).
%F T(n,1) = n.
%F T(n,2) = n for n > 3.
%F For n > 4, T(n,3) >= n+1 if n is odd and T(n,3) >= n+2 if n is even.
%F Conjecture: For n > 4, T(n,3) = n+1 if n is odd and T(n,3) = n+2 if n is even.
%F Conjecture: If n is even, then T(n,k) <= n+k-2 for k >= 2n.
%F (End)
%e The array begins:
%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
%e 2, 2, 4, 4, 5, 6, 7, 8, 9, 10, ...
%e 3, 4, 4, 5, 6, 8, 8, 10, 10, 12, ...
%e 4, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
%e 5, 5, 6, 7, 8, 9, 10, 12, 12, 14, ...
%e 6, 6, 8, 8, 9, 10, 11, 12, 12, 14, ...
%e 7, 7, 8, 9, 10, 11, 12, 13, 14, 16, ...
%e 8, 8, 10, 10, 12, 12, 13, 14, 16, 16, ...
%e 9, 9, 10, 11, 12, 12, 14, 16, 16, 18, ...
%e 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, ...
%e ...
%e As a triangle:
%e 1,
%e 2, 2,
%e 3, 2, 3,
%e 4, 4, 4, 4,
%e 5, 4, 4, 4, 5,
%e 6, 5, 5, 5, 5, 6,
%e 7, 6, 6, 6, 6, 6, 7,
%e 8, 7, 8, 7, 7, 8, 7, 8,
%e 9, 8, 8, 8, 8, 8, 8, 8, 9,
%e 10, 9, 10, 9, 9, 9, 9, 10, 9, 10,
%e ...
%e Illustration for T(2,3) = 4:
%e XOX
%e XOX
%e Illustration for T(2,5) = 5:
%e XXXXX
%e OOOOO
%e Illustration for T(3,5) = 6 (this left edge + top edge construction - or a slight modification of it - works in many cases):
%e OXXXX
%e XOOOO
%e XOOOO
%e Illustration for T(3,6) = 8:
%e XXOOXX
%e OOOOOO
%e XXOOXX
%e Illustration for T(3,8) = 10:
%e OXXXXXXO
%e XOOOOOOX
%e XOOOOOOX
%e Illustration for T(6,9) = 12:
%e OXOOOOOOX
%e OOXXXXXXO
%e OOOOOOOOO
%e OXOOOOOOX
%e OXOOOOOOX
%e OOOOOOOOO
%e From _Bob Selcoe_, Apr 24 2016 (Start)
%e Two symmetric illustrations for T(6,9) = 12:
%e Grid 1:
%e X X O O O O O X X
%e O O O O O O O O O
%e O O O O O O O O O
%e O X X X O X X X O
%e X O O O O O O O X
%e O O O O O O O O O
%e Grid 2:
%e X O O O O O O O X
%e X O O O O O O O X
%e O O O O O O O O O
%e O X X X O X X X O
%e X O O O O O O O X
%e O O O O O O O O O
%e (Note that a symmetric solution is obtained for T(5,9) = 12 by removing row 6)
%e (End)
%Y Cf. A271910.
%Y Main diagonal is A271907.
%K nonn,tabl,more
%O 1,2
%A _Rob Pratt_ and _N. J. A. Sloane_, Apr 24 2016
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