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A235453 Triangle T(n, k) = Number of non-equivalent (mod D_4) ways to arrange k indistinguishable points on an n X n square grid so that no three of them are collinear. Triangle read by rows.

%I A235453 #16 Feb 16 2025 08:33:21
%S A235453 1,0,1,2,1,1,3,8,13,15,5,1,3,21,70,181,217,142,28,4,6,49,290,1253,
%T A235453 3192,4699,3385,1076,110,5,6,93,867,6044,27041,77970,134353,129929,
%U A235453 62177,12511,717,11,10,171,2266,22302,149217,672506,1958674,3531747,3695848,2068757
%N A235453 Triangle T(n, k) = Number of non-equivalent (mod D_4) ways to arrange k indistinguishable points on an n X n square grid so that no three of them are collinear. Triangle read by rows.
%C A235453 The triangle T(n, k) is irregularly shaped: 1 <= k <= 2n. First row corresponds to n = 1.
%C A235453 Without the restriction "non-equivalent (mod D_4)" the numbers are given by triangle A194193. (But this one is read by antidiagonals!)
%C A235453 T(n, 2n) = A000769(n).
%C A235453 2n is an upper bound on the number of points that can be placed on the grid. For large n, it is conjectured that this bound is not reached (see MathWorld link).
%H A235453 Heinrich Ludwig, <a href="/A235453/b235453.txt">Table of n, a(n) for n = 1..99</a>
%H A235453 Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/no3in/readme.html">Progress in the no-three-in-line problem</a>
%H A235453 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/No-Three-in-a-Line-Problem.html">No-Three-in-a-Line-Problem</a>
%e A235453 Triangle begins
%e A235453 1,  0;
%e A235453 1,  2,   1,    1;
%e A235453 3,  8,  13,   15,     5,     1;
%e A235453 3, 21,  70,  181,   217,   142,     28,      4;
%e A235453 6, 49, 290, 1253,  3192,  4699,   3385,   1076,   110,     5;
%e A235453 6, 93, 867, 6044, 27041, 77970, 134353, 129929, 62177, 12511, 717, 11;
%e A235453 ...
%Y A235453 Cf. A194193, A000938, A000769.
%Y A235453 Column 1 is A008805
%Y A235453 Column 2 is A014409
%Y A235453 Column 3 is A235454
%Y A235453 Column 4 is A235455
%Y A235453 Column 5 is A235456
%Y A235453 Column 6 is A235457
%Y A235453 Column 7 is A235458
%K A235453 nonn,tabf
%O A235453 1,4
%A A235453 _Heinrich Ludwig_, Jan 12 2014