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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. 2

%I

%S 0,0,1,17,105,407,1216,3036,6696,13428,25005,43861,73277,117471,

%T 181880,273268,399960,572076,801825,1103625,1494541,1994387,2626152,

%U 3416300,4395148,5596992,7060737,8830137,10954197,13487527,16490972,20031672

%N 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 Column 3 of A194136.

%H 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 StackExchange, <a href="http://math.stackexchange.com/questions/94684">Number of possible triangles in a given triangle.</a>

%F 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 Some solutions for 3X3X3

%e ....0......1......1......0......1......1......1......0......0......1......0

%e ...1.1....1.0....0.1....1.0....0.0....0.0....1.1....1.1....0.1....0.1....0.1

%e ..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 Cf. A194136.

%K nonn

%O 0,4

%A _R. H. Hardin_, Aug 17 2011

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Last modified September 19 03:31 EDT 2021. Contains 347550 sequences. (Running on oeis4.)