A271914 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.

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, 7, 8, 9, 8, 8, 8, 8, 8, 8, 8, 9, 10, 9, 10, 9, 9, 9, 9, 10, 9, 10

Offset: 1

Author

Rob Pratt and N. J. A. Sloane, Apr 24 2016

Keywords

• nonn
• tabl
• more

Comments

It is conjectured that T(n,k) <= n+k-1.

The array is symmetric: T(n,k) = T(k,n).

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.)

The triangle must have nonzero area (three collinear points don't count as a triangle).

Examples

			The array begins:
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
   2,  2,  4,  4,  5,  6,  7,  8,  9, 10, ...
   3,  4,  4,  5,  6,  8,  8, 10, 10, 12, ...
   4,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
   5,  5,  6,  7,  8,  9, 10, 12, 12, 14, ...
   6,  6,  8,  8,  9, 10, 11, 12, 12, 14, ...
   7,  7,  8,  9, 10, 11, 12, 13, 14, 16, ...
   8,  8, 10, 10, 12, 12, 13, 14, 16, 16, ...
   9,  9, 10, 11, 12, 12, 14, 16, 16, 18, ...
  10, 10, 12, 12, 14, 14, 16, 16, 18, 18, ...
  ...
As a triangle:
   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,  7,  8,
   9,  8,  8,  8,  8,  8,  8,  8,  9,
  10,  9, 10,  9,  9,  9,  9, 10,  9, 10,
  ...
Illustration for T(2,3) = 4:
XOX
XOX
Illustration for T(2,5) = 5:
XXXXX
OOOOO
Illustration for T(3,5) = 6 (this left edge + top edge construction - or a slight modification of it - works in many cases):
OXXXX
XOOOO
XOOOO
Illustration for T(3,6) = 8:
XXOOXX
OOOOOO
XXOOXX
Illustration for T(3,8) = 10:
OXXXXXXO
XOOOOOOX
XOOOOOOX
Illustration for T(6,9) = 12:
OXOOOOOOX
OOXXXXXXO
OOOOOOOOO
OXOOOOOOX
OXOOOOOOX
OOOOOOOOO
From _Bob Selcoe_, Apr 24 2016 (Start)
Two symmetric illustrations for T(6,9) = 12:
Grid 1:
X X O O O O O X X
O O O O O O O O O
O O O O O O O O O
O X X X O X X X O
X O O O O O O O X
O O O O O O O O O
Grid 2:
X O O O O O O O X
X O O O O O O O X
O O O O O O O O O
O X X X O X X X O
X O O O O O O O X
O O O O O O O O O
(Note that a symmetric solution is obtained for T(5,9) = 12 by removing row 6)
(End)
		

Links

• Rob Pratt, Complete list of examples where T(n,k) != n+k-2 for 10 >= n >= k >= 2. Note T(9,6) = T(6,9) = 12, which is n+k-3.

Crossrefs

Cf. A271910.

Main diagonal is A271907.

Formula

From Chai Wah Wu, Nov 30 2016: (Start)

T(n,k) >= max(n,k).

T(n,max(k,m)) <= T(n,k+m) <= T(n,k) + T(n,m).

T(n,1) = n.

T(n,2) = n for n > 3.

For n > 4, T(n,3) >= n+1 if n is odd and T(n,3) >= n+2 if n is even.

Conjecture: For n > 4, T(n,3) = n+1 if n is odd and T(n,3) = n+2 if n is even.

Conjecture: If n is even, then T(n,k) <= n+k-2 for k >= 2n.

(End)