A077435 Number of right triangles whose vertices are lattice points in {1,2,...,n} X {1,2,...,n}.

0, 4, 44, 200, 596, 1444, 2960, 5520, 9496, 15332, 23596, 34936, 50020, 69732, 94816, 126176, 164960, 212372, 269620, 337960, 418716, 513444, 623736, 751152, 897776, 1065220, 1255460, 1470680, 1713052, 1984564, 2288304, 2626160, 3000960, 3415124, 3871108

Offset: 1

Author

John W. Layman, Nov 30 2002

Keywords

• nonn

Comments

It would be nice to have a formula. - N. J. A. Sloane, Jun 29 2016

Examples

			For n=2 if the four points are labeled
  ab
  cd
then the right triangles are abc, abd, acd, bcd, so a(2)=4.
For n=3, label the points
  abc
  def
  ghi
The right triangles are: abd (4*4 ways), acg (4 ways), acd and adf (8 ways each), ace and dbf (4 ways each), for a total of a(3) = 44. - _N. J. A. Sloane_, Jun 30 2016
		

Links

• Lars Blomberg, Table of n, a(n) for n = 1..10000 (the first 184 terms from R. H. Hardin)

Crossrefs

Cf. A187452, A279433.

Formula

Place all bounding boxes of A279433 that will fit into the n X n grid in all possible positions, and the proper rectangles in two orientations: a(n) = Sum_{i=1..n} Sum_{j=1..i} k * (n-i+1) * (n-j+1) * A279433(i,j) where k=1 when i=j and k=2 otherwise. - Lars Blomberg, Mar 01 2017

Extensions

a(1) corrected by Lars Blomberg, Mar 01 2017