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A045996 Number of triangles in an n X n grid (or geoplane). 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..28.

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(Sum((n - k + 1)*(n - l + 1)*gcd(k - 1, l - 1), k, 2, n), l, 2, n).

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

f[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[f, 28] (* Robert G. Wilson v, May 23 2010 *)

CROSSREFS

Sequence in context: A317903 A101718 A094160 * A190395 A240281 A114453

Adjacent sequences:  A045993 A045994 A045995 * A045997 A045998 A045999

KEYWORD

nice,nonn,easy

AUTHOR

Ignacio Larrosa CaƱestro

STATUS

approved

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Last modified December 19 10:50 EST 2018. Contains 318246 sequences. (Running on oeis4.)