A045996 Number of triangles in an n X n grid (or geoplane).

0, 4, 76, 516, 2148, 6768, 17600, 40120, 82608, 157252, 280988, 477012, 775172, 1214768, 1844512, 2725000, 3930384, 5550844, 7692300, 10482124, 14066996, 18619128, 24337056, 31449200, 40212160, 50921316, 63907468, 79542108

Offset

1,2

Comments

The triangles must have nonzero area -- their vertices must not be collinear.

The degenerate (i.e., collinear) triangles are counted in A000938. The 1000-term b-file there could be used to produce a 1000-term b-file for the present sequence. - N. J. A. Sloane, Jun 19 2020

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

I. L. Canestro, Checkerboard, sci.math 22 Oct 2000 [broken link]

I. L. Canestro, Checkerboard, sci.math 22 Oct 2000 [Cached copy]

Formula

a(n) = ((n-1)^2*n^2*(n+1)^2)/6 - 2*Sum_{m=2..n} Sum_{k=2..n} (n-k+1)*(n-m+1)*gcd(k-1, m-1).

a(n) = binomial(n^2,3) - A000938(n). - R. J. Mathar, May 21 2010

Example

a(2)=4 because 4 isosceles right triangles can be placed on a 2 X 2 grid.

Mathematica

a[n_] := ((n - 1)^2*n^2*(n + 1)^2)/6 - 2*Sum[(n - k + 1)*(n - l + 1)*GCD[k - 1, l - 1], {k, 2, n}, {l, 2, n}]; Array[a, 28] (* Robert G. Wilson v, May 23 2010 *)

Crossrefs

Cf. A000938.

Sequence in context: A317903 A101718 A094160 * A190395 A240281 A114453

Adjacent sequences: A045993 A045994 A045995 * A045997 A045998 A045999

Keyword

nice,nonn,easy

Author

Ignacio Larrosa Cañestro

Status

approved