

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
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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 1000term bfile there could be used to produce a 1000term bfile for the present sequence.  N. J. A. Sloane, Jun 19 2020


LINKS

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) = ((n1)^2*n^2*(n+1)^2)/6  2*Sum_{m=2..n} Sum_{k=2..n} (nk+1)*(nm+1)*gcd(k1, m1).


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



KEYWORD

nice,nonn,easy


AUTHOR



STATUS

approved



