A194193 -id:A194193 - cp's OEIS frontend

A279445 Triangle read by rows: T(n, k) is the number of ways to place k points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

1, 1, 1, 4, 6, 4, 1, 1, 9, 36, 78, 90, 45, 6, 1, 16, 120, 528, 1428, 2304, 2040, 816, 90, 1, 25, 300, 2200, 10600, 34020, 71400, 93000, 67950, 22650, 2040, 1, 36, 630, 6900, 51525, 270720, 1005720, 2602800, 4531950, 4987800, 3110940, 888840, 67950, 1, 49, 1176, 17934

Offset: 1

Heinrich Ludwig, Dec 17 2016

Length of n-th row is A272651(n) + 1, where A272651(n) is the maximal number of points to be placed under the condition mentioned.
Rotations and reflections of placements are counted. If they are to be ignored, see A279453.
For condition "no more than 2 points on a straight line at any angle", see A194193 (but that one is read by antidiagonals).

			The table begins with T(1, 0):
1  1
1  4   6    4     1
1  9  36   78    90    45     6
1 16 120  528  1428  2304  2040   816    90
1 25 300 2200 10600 34020 71400 93000 67950 22650 2040
...
T(3, 2) = 36 because there are 36 ways to place 2 points on a 3 X 3 square grid so that no more than 2 points are on a vertical or horizontal straight line.

Heinrich Ludwig, Table of n, a(n) for n = 1..109

Row sums give A197458.
Columns 2..10: A000290, A083374, A279437, A279438, A279439, A279440, A279441, A279442, A279443.
Diagonal T(n, n) is A279444.
Cf. A279453, A194193.

A175383 Number of complete quadrangles on an n X n grid (or geoplane).

Original entry on oeis.org

0, 1, 78, 1278, 9498, 47331, 175952, 545764, 1461672, 3507553, 7701638, 15773526, 30375194, 55695587, 97777392, 165310348, 270478344, 430196181, 666685134, 1010083690, 1498720098, 2182544223

Offset: 1

Martin Renner, Apr 19 2011

nonn

A complete quadrangle is a set of four points, no three collinear, and the six lines which join them.

Number of ways to arrange 4 indistinguishable points on an n X n square grid so that no three points are collinear at any angle. Column 4 of A194193. - R. H. Hardin, Aug 18 2011

			From _R. H. Hardin_, Aug 18 2011: (Start)
Some solutions for 3 X 3:
  0 1 1   1 1 0   1 0 1   0 1 1   0 0 0   1 1 0   1 1 0
  1 0 0   0 0 0   1 0 0   1 1 0   1 1 0   0 0 1   1 0 0
  1 0 0   1 0 1   0 0 1   0 0 0   0 1 1   0 1 0   0 1 0
(End)

R. H. Hardin, Table of n, a(n) for n = 1..55
Eric Weisstein's World of Mathematics, Complete Quadrangle.

a(n) = A189345(n) - A189346(n) - A178256(n).

a(n) = (1/3)*A189412(n) + A189413(n).

a(6)-a(22) from Nathaniel Johnston, Apr 25 2011

a(7)-a(22) corrected by Nathaniel Johnston, based on another correction by Michal Forišek, Sep 06 2011

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.

Original entry on oeis.org

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

Offset: 1

Heinrich Ludwig, Jan 12 2014

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).

			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;
...

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

Cf. A194193, A000938, A000769.
Column 1 is A008805
Column 2 is A014409
Column 3 is A235454
Column 4 is A235455
Column 5 is A235456
Column 6 is A235457
Column 7 is A235458

A194190 Number of ways to arrange 5 indistinguishable points on an n X n square grid so that no three points are collinear at any angle.

Original entry on oeis.org

0, 0, 28, 1668, 25052, 215448, 1189868, 5199888, 18520572, 56978440, 155627304, 388897892, 894254904, 1932504496, 3945470564, 7669533756, 14291010972, 25694009628, 44662697948, 75451394832, 124066723008, 199190308172

Offset: 1

R. H. Hardin, Aug 18 2011

nonn

Column 5 of A194193.

			Some solutions for 3 X 3:
..0..1..0....1..0..1....0..1..1....1..1..0....1..1..0....1..1..0....1..0..1
..0..1..1....1..1..0....1..0..0....1..0..1....1..0..0....0..0..1....1..0..0
..1..0..1....0..1..0....0..1..1....0..0..1....0..1..1....1..1..0....0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..24

A194191 Number of ways to arrange 6 indistinguishable points on an n X n square grid so that no three points are collinear at any angle.

Original entry on oeis.org

0, 0, 2, 998, 36698, 620210, 5367308, 34678364, 169259212, 682686652, 2356999994, 7294368210, 20227526910, 52008171998, 124422857864, 279767468172, 596674510744, 1218556387684, 2385034544810, 4509309201242

Offset: 1

R. H. Hardin, Aug 18 2011

nonn

Column 6 of A194193.

			Some solutions for 4 X 4:
..1..0..1..0....0..1..1..0....1..0..0..1....0..1..0..0....0..1..0..1
..0..0..0..1....1..0..0..0....0..0..1..1....0..0..1..1....0..0..0..1
..0..1..0..0....0..1..1..0....0..0..0..0....0..0..1..0....0..0..1..0
..0..0..1..1....1..0..0..0....0..1..1..0....1..0..0..1....1..0..1..0

A194192 Number of ways to arrange 7 indistinguishable points on an n X n square grid so that no three points are collinear at any angle.

Original entry on oeis.org

0, 0, 0, 204, 26700, 1073076, 15657764, 159413700, 1108580092, 6030207624, 26852315940, 104865006648, 354993232192, 1098793355508, 3109294280324

Offset: 1

R. H. Hardin, Aug 18 2011

nonn

Column 7 of A194193.

			Some solutions for 4 X 4:
..0..0..1..1....1..0..1..0....0..0..1..0....0..0..0..1....1..0..1..0
..0..0..0..1....0..0..0..1....0..0..1..1....1..0..1..0....0..1..0..0
..1..1..0..0....0..1..0..1....1..1..0..0....1..0..1..0....0..1..0..1
..0..1..1..0....0..1..1..0....0..1..0..1....0..1..0..1....1..0..1..0

cp's OEIS Frontend

A279445 Triangle read by rows: T(n, k) is the number of ways to place k points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

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A175383 Number of complete quadrangles on an n X n grid (or geoplane).

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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.

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A194190 Number of ways to arrange 5 indistinguishable points on an n X n square grid so that no three points are collinear at any angle.

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A194191 Number of ways to arrange 6 indistinguishable points on an n X n square grid so that no three points are collinear at any angle.

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A194192 Number of ways to arrange 7 indistinguishable points on an n X n square grid so that no three points are collinear at any angle.

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