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A194131 Number of ways to arrange 3 indistinguishable points on an n X n X n triangular grid so that no three points are collinear at any angle.

%I A194131 #23 Dec 09 2017 03:57:09
%S A194131 0,0,1,17,105,407,1216,3036,6696,13428,25005,43861,73277,117471,
%T A194131 181880,273268,399960,572076,801825,1103625,1494541,1994387,2626152,
%U A194131 3416300,4395148,5596992,7060737,8830137,10954197,13487527,16490972,20031672
%N A194131 Number of ways to arrange 3 indistinguishable points on an n X n X n triangular grid so that no three points are collinear at any angle.
%C A194131 Column 3 of A194136.
%H A194131 N. J. A. Sloane, <a href="/A194131/b194131.txt">Table of n, a(n) for n = 0..500</a>, Mar 20 2016 [First 196 terms from _R. H. Hardin_]
%H A194131 StackExchange, <a href="http://math.stackexchange.com/questions/94684">Number of possible triangles in a given triangle.</a>
%F A194131 a(n) = ((n^2+n+2)/2) * binomial(n+2,4) - (3/2) * Sum_{k=2..n} (n-k+1) * (n-k+2) * Sum_{m=2..k} gcd(k-1,m-1). - _David Bevan_, Jan 01 2012
%e A194131 Some solutions for 3X3X3
%e A194131 ....0......1......1......0......1......1......1......0......0......1......0
%e A194131 ...1.1....1.0....0.1....1.0....0.0....0.0....1.1....1.1....0.1....0.1....0.1
%e A194131 ..0.0.1..0.0.1..1.0.0..1.1.0..0.1.1..1.0.1..0.0.0..0.1.0..1.1.0..0.1.0..1.0.1
%Y A194131 Cf. A194136.
%K A194131 nonn
%O A194131 0,4
%A A194131 _R. H. Hardin_, Aug 17 2011