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 A045996 Number of triangles in an n X n grid (or geoplane). 21
 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. 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

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Last modified December 6 16:13 EST 2023. Contains 367612 sequences. (Running on oeis4.)