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A194136 T(n,k) = number of ways to arrange k indistinguishable points on an n X n X n triangular grid so that no three points are collinear at any angle.

%I A194136 #12 May 12 2025 12:00:44
%S A194136 1,0,3,0,3,6,0,1,15,10,0,0,17,45,15,0,0,6,105,105,21,0,0,0,114,407,
%T A194136 210,28,0,0,0,39,843,1216,378,36,0,0,0,1,792,4122,3036,630,45,0,0,0,0,
%U A194136 244,7587,14988,6696,990,55,0,0,0,0,9,6480,43836,45414,13428,1485,66,0,0,0,0
%N A194136 T(n,k) = number of ways to arrange k indistinguishable points on an n X n X n triangular grid so that no three points are collinear at any angle.
%C A194136 Table starts
%C A194136 ...1....0......0.......0.........0..........0...........0............0
%C A194136 ...3....3......1.......0.........0..........0...........0............0
%C A194136 ...6...15.....17.......6.........0..........0...........0............0
%C A194136 ..10...45....105.....114........39..........1...........0............0
%C A194136 ..15..105....407.....843.......792........244...........9............0
%C A194136 ..21..210...1216....4122......7587.......6480........1875...........78
%C A194136 ..28..378...3036...14988.....43836......69798.......52323........14268
%C A194136 ..36..630...6696...45414....194013.....496198......695616.......464934
%C A194136 ..45..990..13428..119340....696765....2595897.....5840088......7278867
%C A194136 ..55.1485..25005..281442...2145687...10912452....35715529.....71089536
%C A194136 ..66.2145..43861..608616...5851044...38739354...172520643....496946172
%C A194136 ..78.3003..73277.1228812..14546412..121694240...708871152...2796515883
%C A194136 ..91.4095.117471.2338779..33347130..342722071..2517687856..12966188538
%C A194136 .105.5460.181880.4240284..71662911..887407361..8023634766..52262929401
%C A194136 .120.7140.273268.7371414.145616964.2136513884.23292994812.187041426756
%H A194136 R. H. Hardin, <a href="/A194136/b194136.txt">Table of n, a(n) for n = 1..267</a>
%H A194136 S. V. Ullas Chandran, Sandi Klavžar, and James Tuite, <a href="https://arxiv.org/abs/2501.19385">The General Position Problem: A Survey</a>, arXiv:2501.19385 [math.CO], 2025. See p. 4.
%e A194136 Some solutions for n=4, k=4:
%e A194136 .....1........0........1........0........0........0........0........0
%e A194136 ....0.0......1.1......1.0......0.1......1.1......1.0......1.1......0.1
%e A194136 ...0.1.0....0.0.1....0.0.0....1.0.0....1.0.0....1.0.1....1.1.0....1.1.0
%e A194136 ..1.0.0.1..1.0.0.0..0.1.1.0..0.1.0.1..0.0.1.0..0.0.1.0..0.0.0.0..0.0.0.1
%Y A194136 Column 1 is A000217.
%Y A194136 Column 2 is A050534.
%K A194136 nonn,tabl
%O A194136 1,3
%A A194136 _R. H. Hardin_, Aug 17 2011