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

1,3

Comments

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

Links

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

Example

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.

Sequence in context: A167350 A215669 A244310 * A156457 A058348 A184946

Adjacent sequences: A334702 A334703 A334704 * A334706 A334707 A334708

Keyword

nonn,tabl

Author

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

Extensions

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

Status

approved