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.

0, 0, 1, 17, 105, 407, 1216, 3036, 6696, 13428, 25005, 43861, 73277, 117471, 181880, 273268, 399960, 572076, 801825, 1103625, 1494541, 1994387, 2626152, 3416300, 4395148, 5596992, 7060737, 8830137, 10954197, 13487527, 16490972, 20031672

Offset: 0

Author

R. H. Hardin, Aug 17 2011

Keywords

• nonn

Comments

Column 3 of A194136.

Examples

			Some solutions for 3X3X3
....0......1......1......0......1......1......1......0......0......1......0
...1.1....1.0....0.1....1.0....0.0....0.0....1.1....1.1....0.1....0.1....0.1
..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
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..500, Mar 20 2016 [First 196 terms from _R. H. Hardin_]
• StackExchange, Number of possible triangles in a given triangle.

Crossrefs

Cf. A194136.

Formula

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