This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A271914 #63 Dec 01 2016 01:33:04 %S A271914 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 A271914 7,8,9,8,8,8,8,8,8,8,9,10,9,10,9,9,9,9,10,9,10 %N 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. %C A271914 It is conjectured that T(n,k) <= n+k-1. %C A271914 The array is symmetric: T(n,k) = T(k,n). %C A271914 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 A271914 The triangle must have nonzero area (three collinear points don't count as a triangle). %H A271914 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 A271914 From _Chai Wah Wu_, Nov 30 2016: (Start) %F A271914 T(n,k) >= max(n,k). %F A271914 T(n,max(k,m)) <= T(n,k+m) <= T(n,k) + T(n,m). %F A271914 T(n,1) = n. %F A271914 T(n,2) = n for n > 3. %F A271914 For n > 4, T(n,3) >= n+1 if n is odd and T(n,3) >= n+2 if n is even. %F A271914 Conjecture: For n > 4, T(n,3) = n+1 if n is odd and T(n,3) = n+2 if n is even. %F A271914 Conjecture: If n is even, then T(n,k) <= n+k-2 for k >= 2n. %F A271914 (End) %e A271914 The array begins: %e A271914 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... %e A271914 2, 2, 4, 4, 5, 6, 7, 8, 9, 10, ... %e A271914 3, 4, 4, 5, 6, 8, 8, 10, 10, 12, ... %e A271914 4, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... %e A271914 5, 5, 6, 7, 8, 9, 10, 12, 12, 14, ... %e A271914 6, 6, 8, 8, 9, 10, 11, 12, 12, 14, ... %e A271914 7, 7, 8, 9, 10, 11, 12, 13, 14, 16, ... %e A271914 8, 8, 10, 10, 12, 12, 13, 14, 16, 16, ... %e A271914 9, 9, 10, 11, 12, 12, 14, 16, 16, 18, ... %e A271914 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, ... %e A271914 ... %e A271914 As a triangle: %e A271914 1, %e A271914 2, 2, %e A271914 3, 2, 3, %e A271914 4, 4, 4, 4, %e A271914 5, 4, 4, 4, 5, %e A271914 6, 5, 5, 5, 5, 6, %e A271914 7, 6, 6, 6, 6, 6, 7, %e A271914 8, 7, 8, 7, 7, 8, 7, 8, %e A271914 9, 8, 8, 8, 8, 8, 8, 8, 9, %e A271914 10, 9, 10, 9, 9, 9, 9, 10, 9, 10, %e A271914 ... %e A271914 Illustration for T(2,3) = 4: %e A271914 XOX %e A271914 XOX %e A271914 Illustration for T(2,5) = 5: %e A271914 XXXXX %e A271914 OOOOO %e A271914 Illustration for T(3,5) = 6 (this left edge + top edge construction - or a slight modification of it - works in many cases): %e A271914 OXXXX %e A271914 XOOOO %e A271914 XOOOO %e A271914 Illustration for T(3,6) = 8: %e A271914 XXOOXX %e A271914 OOOOOO %e A271914 XXOOXX %e A271914 Illustration for T(3,8) = 10: %e A271914 OXXXXXXO %e A271914 XOOOOOOX %e A271914 XOOOOOOX %e A271914 Illustration for T(6,9) = 12: %e A271914 OXOOOOOOX %e A271914 OOXXXXXXO %e A271914 OOOOOOOOO %e A271914 OXOOOOOOX %e A271914 OXOOOOOOX %e A271914 OOOOOOOOO %e A271914 From _Bob Selcoe_, Apr 24 2016 (Start) %e A271914 Two symmetric illustrations for T(6,9) = 12: %e A271914 Grid 1: %e A271914 X X O O O O O X X %e A271914 O O O O O O O O O %e A271914 O O O O O O O O O %e A271914 O X X X O X X X O %e A271914 X O O O O O O O X %e A271914 O O O O O O O O O %e A271914 Grid 2: %e A271914 X O O O O O O O X %e A271914 X O O O O O O O X %e A271914 O O O O O O O O O %e A271914 O X X X O X X X O %e A271914 X O O O O O O O X %e A271914 O O O O O O O O O %e A271914 (Note that a symmetric solution is obtained for T(5,9) = 12 by removing row 6) %e A271914 (End) %Y A271914 Cf. A271910. %Y A271914 Main diagonal is A271907. %K A271914 nonn,tabl,more %O A271914 1,2 %A A271914 _Rob Pratt_ and _N. J. A. Sloane_, Apr 24 2016