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

0,4

Comments

Column 3 of A194136.

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.

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

Example

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

Crossrefs

Cf. A194136.

Sequence in context: A242316 A329386 A095785 * A194475 A164745 A221938

Adjacent sequences: A194128 A194129 A194130 * A194132 A194133 A194134

Keyword

nonn

Author

R. H. Hardin, Aug 17 2011

Status

approved