A334705 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ways to choose three points from an n X k grid of points which are the vertices of a triangle of nonzero area.

0, 0, 4, 0, 18, 76, 0, 48, 200, 516, 0, 100, 412, 1056, 2148, 0, 180, 738, 1884, 3820, 6768, 0, 294, 1200, 3052, 6176, 10922, 17600, 0, 448, 1824, 4628, 9352, 16516, 26588, 40120, 0, 648, 2632, 6668, 13456, 23740, 38192, 57588, 82608, 0, 900, 3650, 9232, 18612, 32812, 52758, 79508, 114000, 157252

Offset: 1

Author

N. J. A. Sloane, Jun 13 2020

Keywords

• nonn
• tabl

Comments

It follows from the definitions that T(n,k) + A334704(n,k) = A334703(n,k) for 1 <= k <= n.

Examples

			Triangle begins:
0,
0, 4,
0, 18, 76,
0, 48, 200, 516,
0, 100, 412, 1056, 2148,
0, 180, 738, 1884, 3820, 6768,
0, 294, 1200, 3052, 6176, 10922, 17600,
0, 448, 1824, 4628, 9352, 16516, 26588, 40120,
0, 648, 2632, 6668, 13456, 23740, 38192, 57588, 82608,
0, 900, 3650, 9232, 18612, 32812, 52758, 79508, 114000, 157252,
0, 1210, 4900, 12380, 24940, 43934, 70608, 106364, 152456, 210234, 280988,
...
This is the lower half of a symmetric array. The full symmetric array begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 4, 18, 48, 100, 180, 294, 448, 648, 900, 1210, 1584, ...
0, 18, 76, 200, 412, 738, 1200, 1824, 2632, 3650, 4900, 6408, ...
0, 48, 200, 516, 1056, 1884, 3052, 4628, 6668, 9232, 12380, 16176, ...
0, 100, 412, 1056, 2148, 3820, 6176, 9352, 13456, 18612, 24940, 32568, ...
0, 180, 738, 1884, 3820, 6768, 10922, 16516, 23740, 32812, 43934, 57336, ...
0, 294, 1200, 3052, 6176, 10922, 17600, 26588, 38192, 52758, 70608, 92112, ...
0, 448, 1824, 4628, 9352, 16516, 26588, 40120, 57588, 79508, 106364, 138708, ...
0, 648, 2632, 6668, 13456, 23740, 38192, 57588, 82608, 114000, 152456, 198760, ...
0, 900, 3650, 9232, 18612, 32812, 52758, 79508, 114000, 157252, 210234, 274016 , ...
0, 1210, 4900, 12380, 24940, 43934, 70608, 106364, 152456, 210234, 280988, 366152, ...
...
		

Crossrefs

This is a companion to the triangles A334703 and A334704.

Rows (or columns) 2,3,4,5 of the full array are A045991, A262402, A296367, A334707. The main diagonal is A045996.

Extensions

Rows 6 onwards from Tom Duff (see the Duff link in A334704). - N. J. A. Sloane, Jun 19 2020