A235453 - OEIS

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A235453

Triangle T(n, k) = Number of non-equivalent (mod D_4) ways to arrange k indistinguishable points on an n X n square grid so that no three of them are collinear. Triangle read by rows.

1, 0, 1, 2, 1, 1, 3, 8, 13, 15, 5, 1, 3, 21, 70, 181, 217, 142, 28, 4, 6, 49, 290, 1253, 3192, 4699, 3385, 1076, 110, 5, 6, 93, 867, 6044, 27041, 77970, 134353, 129929, 62177, 12511, 717, 11, 10, 171, 2266, 22302, 149217, 672506, 1958674, 3531747, 3695848, 2068757

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

The triangle T(n, k) is irregularly shaped: 1 <= k <= 2n. First row corresponds to n = 1.

Without the restriction "non-equivalent (mod D_4)" the numbers are given by triangle A194193. (But this one is read by antidiagonals!)

T(n, 2n) = A000769(n).

2n is an upper bound on the number of points that can be placed on the grid. For large n, it is conjectured that this bound is not reached (see MathWorld link).

LINKS

Heinrich Ludwig, Table of n, a(n) for n = 1..99

Achim Flammenkamp, Progress in the no-three-in-line problem

Eric Weisstein's World of Mathematics, No-Three-in-a-Line-Problem

EXAMPLE

Triangle begins

1, 0;

1, 2, 1, 1;

3, 8, 13, 15, 5, 1;

3, 21, 70, 181, 217, 142, 28, 4;

6, 49, 290, 1253, 3192, 4699, 3385, 1076, 110, 5;

6, 93, 867, 6044, 27041, 77970, 134353, 129929, 62177, 12511, 717, 11;