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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

LINKS

Table of n, a(n) for n=1..55.

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.

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)

EXAMPLE

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)

CROSSREFS

Cf. A271910.

Main diagonal is A271907.

Sequence in context: A127095 A249871 A074712 * A087735 A277194 A172151

Adjacent sequences:  A271911 A271912 A271913 * A271915 A271916 A271917

KEYWORD

nonn,tabl,more

AUTHOR

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

STATUS

approved

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Last modified October 22 06:02 EDT 2018. Contains 316432 sequences. (Running on oeis4.)