A320539 (1/2) * number of ways to select 3 distinct collinear points from a rectangle of grid points with side lengths j and k, written as triangle T(j,k), j<=k.

0, 1, 4, 4, 10, 22, 10, 21, 42, 76, 20, 39, 70, 120, 186, 35, 65, 112, 184, 279, 412, 56, 100, 166, 264, 390, 566, 772, 84, 146, 236, 367, 532, 759, 1026, 1356, 120, 205, 324, 494, 704, 991, 1326, 1740, 2224, 165, 278, 432, 647, 913, 1271, 1686, 2196, 2793, 3496

Offset

1,3

Comments

Permutations of the 3 points are not counted separately.

Links

Table of n, a(n) for n=1..55.

Example

The triangle begins:
    0
    1    4
    4   10   22
   10   21   42   76
   20   39   70  120  186
   35   65  112  184  279  412
   56  100  166  264  390  566  772
.
a(2) = T(1,2) = 1, because the grid points on the two longer sides of the rectangle are collinear: (0,0) (0,1) (0,2) and (1,0) (1,1) (2,2).
a(3) = T(2,2) = 4, because there are 8 triples of collinear points:
  (0,0) (0,1) (0,2),
  (0,0) (1,0) (2,0),
  (0,0) (1,1) (2,2),
  (0,1) (1,1) (2,1),
  (0,2) (1,1) (2,0),
  (0,2) (1,2) (2,2),
  (1,0) (1,1) (1,2),
  (2,0) (2,1) (2,2).

Crossrefs

A000292, A320540, A320541, A320543.

Sequence in context: A178820 A284784 A219803 * A145598 A320392 A117881

Adjacent sequences: A320536 A320537 A320538 * A320540 A320541 A320542

Keyword

nonn,tabl

Author

Hugo Pfoertner, Oct 15 2018

Status

approved