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

1,5

Comments

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

Links

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

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

Example

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, ...
...

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.

Sequence in context: A365478 A271836 A349823 * A021355 A323986 A334447

Adjacent sequences: A334701 A334702 A334703 * A334705 A334706 A334707

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jun 13 2020.

Extensions

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

Status

approved