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A045996
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Number of triangles in an n X n grid (or geoplane).
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21
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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I. L. Canestro, Checkerboard, sci.math 22 Oct 2000 [broken link]
I. L. Canestro, Checkerboard, sci.math 22 Oct 2000 [Cached copy]
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FORMULA
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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).
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EXAMPLE
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a(2)=4 because 4 isosceles right triangles can be placed on a 2 X 2 grid.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nice,nonn,easy
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AUTHOR
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STATUS
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approved
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