A334704 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ways to choose three collinear points from an n X k grid of points.

0, 0, 0, 1, 2, 8, 4, 8, 20, 44, 10, 20, 43, 84, 152, 20, 40, 78, 140, 240, 372, 35, 70, 130, 224, 369, 558, 824, 56, 112, 200, 332, 528, 780, 1132, 1544, 84, 168, 293, 472, 734, 1064, 1519, 2052, 2712, 120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448

Offset: 1

Author

N. J. A. Sloane, Jun 13 2020

Keywords

• nonn
• tabl

Comments

It follows from the definitions that T(n,k) + A334705(n,k) = A334703(n,k) for 1 <= k <= n.

Examples

			Triangle begins:
0,
0, 0,
1, 2, 8,
4, 8, 20, 44,
10, 20, 43, 84, 152,
20, 40, 78, 140, 240, 372,
35, 70, 130, 224, 369, 558, 824,
56, 112, 200, 332, 528, 780, 1132, 1544,
84, 168, 293, 472, 734, 1064, 1519, 2052, 2712,
120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448,
165, 330, 556, 864, 1295, 1826, 2542, 3372, 4393, 5586, 6992,
...
This is the lower half of a symmetric array. The full symmetric array begins:
0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ...
0, 0, 2, 8, 20, 40, 70, 112, 168, 240, 330, 440, ...
1, 2, 8, 20, 43, 78, 130, 200, 293, 410, 556, 732, ...
4, 8, 20, 44, 84, 140, 224, 332, 472, 648, 864, 1120, ...
10, 20, 43, 84, 152, 240, 369, 528, 734, 988, 1295, 1652, ...
20, 40, 78, 140, 240, 372, 558, 780, 1064, 1408, 1826, 2304, ...
35, 70, 130, 224, 369, 558, 824, 1132, 1519, 1982, 2542, 3172, ...
56, 112, 200, 332, 528, 780, 1132, 1544, 2052, 2652, 3372, 4172, ...
84, 168, 293, 472, 734, 1064, 1519, 2052, 2712, 3480, 4393, 5396, ...
120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448, 5586, 6824, ...
165, 330, 556, 864, 1295, 1826, 2542, 3372, 4393, 5586, 6992, 8508, ...
220, 440, 732, 1120, 1652, 2304, 3172, 4172, 5396, 6824, 8508, 10332, ...
...
		

Links

• Tom Duff, Values of T(n,k) for grids of size up to 50 X 50

Crossrefs

This is a companion to the triangles A334703 and A334705.

Rows (or columns) 1,2,3,4 of the full array are A000292, A007290, A057566, A334706. The main diagonal is A000938.

Extensions

Rows 6 onwards from Tom Duff. - N. J. A. Sloane, Jun 19 2020