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 A334711 #20 Jun 23 2020 09:12:36 %S A334711 0,0,0,0,1,0,0,9,9,0,0,36,70,36,0,0,100,276,276,100,0,0,225,750,1038, %T A334711 750,225,0,0,441,1677,2788,2788,1677,441,0,0,784,3260,6190,7398,6190, %U A334711 3260,784,0,0,1296,5776,11942,16328,16328,11942,5776,1296,0,0,2025,9508,21062,31396,35727,31396,21062,9508,2025,0 %N A334711 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that they form a convex quadrilateral. %C A334711 Computed by _Tom Duff_, Jun 15 2020 %C A334711 For the limiting probability that the four points form a convex quadrilateral when n and k are large, see the link to Sylvester's Four-Point Problem. Thanks to _Ed Pegg Jr_ for this comment. %H A334711 Tom Duff, <a href="/A334708/a334708_3.txt">Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192</a> %H A334711 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SylvestersFour-PointProblem.html">Sylvester's Four-Point Problem</a>. %e A334711 The initial rows of the array are: %e A334711 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... %e A334711 0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, ... %e A334711 0, 9, 70, 276, 750, 1677, 3260, 5776, 9508, 14825, 22090, 31764, ... %e A334711 0, 36, 276, 1038, 2788, 6190, 11942, 21062, 34586, 53748, 79930, 114760, ... %e A334711 0, 100, 750, 2788, 7398, 16328, 31396, 55244, 90484, 140372, 208490, 299048, ... %e A334711 0, 225, 1677, 6190, 16328, 35727, 68447, 120106, 196338, 304161, 451035, 646116, ... %e A334711 0, 441, 3260, 11942, 31396, 68447, 130768, 229034, 373968, 578777, 857524, 1227572, ... %e A334711 0, 784, 5776, 21062, 55244, 120106, 229034, 400116, 652318, 1008438, 1492870, 2135534, ... %e A334711 0, 1296, 9508, 34586, 90484, 196338, 373968, 652318, 1062016, 1640284, 2426660, 3469356, ... %e A334711 0, 2025, 14825, 53748, 140372, 304161, 578777, 1008438, 1640284, 2531001, 3742053, 5347100, ... %e A334711 ... %e A334711 The initial antidiagonals are: %e A334711 0, %e A334711 0, 0, %e A334711 0, 1, 0, %e A334711 0, 9, 9, 0, %e A334711 0, 36, 70, 36, 0, %e A334711 0, 100, 276, 276, 100, 0, %e A334711 0, 225, 750, 1038, 750, 225, 0, %e A334711 0, 441, 1677, 2788, 2788, 1677, 441, 0, %e A334711 0, 784, 3260, 6190, 7398, 6190, 3260, 784, 0, %e A334711 0, 1296, 5776, 11942, 16328, 16328, 11942, 5776, 1296, 0, %e A334711 0, 2025, 9508, 21062, 31396, 35727, 31396, 21062, 9508, 2025, 0, %e A334711 0, 3025, 14825, 34586, 55244, 68447, 68447, 55244, 34586, 14825, 3025, 0, %e A334711 ... %Y A334711 The main diagonal is A189413. %Y A334711 Triangles A334708, A334709, A334710, A334711 give the counts for the four possible arrangements of four points. %Y A334711 For three points there are just two possible arrangements: see A334704 and A334705. %K A334711 nonn,tabl %O A334711 1,8 %A A334711 _N. J. A. Sloane_, Jun 15 2020