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
KEYWORD
nice,nonn,easy
AUTHOR
STATUS
approved