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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 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: A249871 A330408 A074712 * A087735 A322630 A277194 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 September 20 04:58 EDT 2021. Contains 347577 sequences. (Running on oeis4.)