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 A045996 Number of triangles in an n X n grid (or geoplane). 20

%I

%S 0,4,76,516,2148,6768,17600,40120,82608,157252,280988,477012,775172,

%T 1214768,1844512,2725000,3930384,5550844,7692300,10482124,14066996,

%U 18619128,24337056,31449200,40212160,50921316,63907468,79542108

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

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

%C 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

%H Vaclav Kotesovec, <a href="/A045996/b045996.txt">Table of n, a(n) for n = 1..1000</a>

%H I. L. Canestro, <a href="http://www.math.niu.edu/~rusin/known-math/00_incoming/triangles_count">Checkerboard</a>, sci.math 22 Oct 2000 [broken link]

%H I. L. Canestro, <a href="/A045996/a045996.txt">Checkerboard</a>, sci.math 22 Oct 2000 [Cached copy]

%F 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).

%F a(n) = binomial(n^2,3)- A000938(n). - _R. J. Mathar_, May 21 2010

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

%t 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 *)

%Y Cf. A000938.

%K nice,nonn,easy

%O 1,2

%A _Ignacio Larrosa CaĆ±estro_

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Last modified September 25 17:14 EDT 2021. Contains 347659 sequences. (Running on oeis4.)