A334711 -id:A334711 - cp's OEIS frontend

A189413 Number of convex quadrilaterals on an n X n grid (or geoboard).

0, 1, 70, 1038, 7398, 35727, 130768, 400116, 1062016, 2531001, 5529310, 11272710, 21639022, 39559591, 69283632, 116910052, 190977408, 303286461, 469431366, 710400658, 1053055398, 1532253131, 2192246528, 3088876728, 4290532688, 5882825641, 7969711934, 10677299074, 14156978846, 18591603883, 24195121104

Offset: 1

Martin Renner, Apr 21 2011

nonn

If four points are chosen at random from an n X n grid, the probability that they form a convex quadrilateral approaches 25/36 as n increases, by Sylvester's Four-Point Theorem (see the link). Thanks to Ed Pegg Jr for this comment. - N. J. A. Sloane, Jun 15 2020

Tom Duff, Table of n, a(n) for n = 1..192
Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192.
Nathaniel Johnston, C program for computing terms.
Eric Weisstein's World of Mathematics, Convex Polygon.
Eric Weisstein's World of Mathematics, Quadrilateral.
Eric Weisstein's World of Mathematics, Sylvester's Four-Point Problem.

Cf. A175383, A178256, A181944, A189345, A189412, A189414.

This is the main diagonal of A334711.

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

Terms beyond a(22) from Tom Duff. - N. J. A. Sloane, Jun 23 2020

A334708 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four collinear points from an n X k grid.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 5, 2, 0, 2, 5, 15, 10, 3, 3, 10, 15, 35, 30, 15, 10, 15, 30, 35, 70, 70, 45, 29, 29, 45, 70, 70, 126, 140, 105, 72, 64, 72, 105, 140, 126, 210, 252, 210, 157, 129, 129, 157, 210, 252, 210, 330, 420, 378, 302, 248, 234, 248, 302, 378, 420, 330

Offset: 1

N. J. A. Sloane, Jun 15 2020

Computed by Tom Duff, Jun 15 2020

			The initial rows of the array are:
0, 0, 0, 1, 5, 15, 35, 70, 126, 210, 330, 495, ...
0, 0, 0, 2, 10, 30, 70, 140, 252, 420, 660, 990, ...
0, 0, 0, 3, 15, 45, 105, 210, 378, 630, 990, 1485, ...
1, 2, 3, 10, 29, 72, 157, 302, 531, 874, 1361, 2028, ...
5, 10, 15, 29, 64, 129, 248, 442, 747, 1196, 1825, 2679, ...
15, 30, 45, 72, 129, 234, 405, 666, 1065, 1638, 2439, 3510, ...
35, 70, 105, 157, 248, 405, 660, 1020, 1545, 2276, 3283, 4605, ...
70, 140, 210, 302, 442, 666, 1020, 1524, 2220, 3154, 4412, 6030, ...
126, 252, 378, 531, 747, 1065, 1545, 2220, 3156, 4362, 5940, 7923, ...
210, 420, 630, 874, 1196, 1638, 2276, 3154, 4362, 5928, 7914, 10350, ...
...
The initial antidiagonals are:
0
0, 0
0, 0, 0
1, 0, 0, 1
5, 2, 0, 2, 5
15, 10, 3, 3, 10, 15
35, 30, 15, 10, 15, 30, 35
70, 70, 45, 29, 29, 45, 70, 70
126, 140, 105, 72, 64, 72, 105, 140, 126
210, 252, 210, 157, 129, 129, 157, 210, 252, 210
330, 420, 378, 302, 248, 234, 248, 302, 378, 420, 330
...

Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192

The main diagonal is A178256.
Triangles A334708, A334709, A334710, A334711 give the counts for the four possible arrangements of four points.
For three points there are just two possible arrangements: see A334704 and A334705.

A334709 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that three of them form a triangle of nonzero area and the extra point is strictly inside the triangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 48, 48, 0, 0, 0, 0, 144, 240, 144, 0, 0, 0, 0, 348, 716, 716, 348, 0, 0, 0, 0, 700, 1712, 2100, 1712, 700, 0, 0, 0, 0, 1280, 3404, 4984, 4984, 3404, 1280, 0, 0, 0, 0, 2144, 6176, 9900, 11604, 9900, 6176, 2144, 0, 0, 0, 0, 3400, 10336, 17936, 22936, 22936, 17936, 10336, 3400, 0, 0

Offset: 1

N. J. A. Sloane, Jun 15 2020

Computed by Tom Duff, Jun 15 2020

			The initial rows of the array are:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 8, 48, 144, 348, 700, 1280, 2144, 3400, 5120, 7440, ...
0, 0, 48, 240, 716, 1712, 3404, 6176, 10336, 16288, 24480, 35504, ...
0, 0, 144, 716, 2100, 4984, 9900, 17936, 29924, 47080, 70700, 102460, ...
0, 0, 348, 1712, 4984, 11604, 22936, 41372, 68844, 108132, 161964, 234228, ...
0, 0, 700, 3404, 9900, 22936, 45184, 81320, 135192, 212152, 317492, 458812, ...
0, 0, 1280, 6176, 17936, 41372, 81320, 145648, 241544, 378400, 565636, 816520, ...
0, 0, 2144, 10336, 29924, 68844, 135192, 241544, 399656, 625232, 933808, 1346928, ...
0, 0, 3400, 16288, 47080, 108132, 212152, 378400, 625232, 976552, 1457172, 2100112, ...
...
The initial antidiagonals are:
0,
0, 0,
0, 0, 0,
0, 0, 0, 0,
0, 0, 8, 0, 0,
0, 0, 48, 48, 0, 0,
0, 0, 144, 240, 144, 0, 0,
0, 0, 348, 716, 716, 348, 0, 0,
0, 0, 700, 1712, 2100, 1712, 700, 0, 0,
0, 0, 1280, 3404, 4984, 4984, 3404, 1280, 0, 0,
0, 0, 2144, 6176, 9900, 11604, 9900, 6176, 2144, 0, 0,
0, 0, 3400, 10336, 17936, 22936, 22936, 17936, 10336, 3400, 0, 0,
...

Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192

The main diagonal is A334712.
Triangles A334708, A334709, A334710, A334711 give the counts for the four possible arrangements of four points.
For three points there are just two possible arrangements: see A334704 and A334705.

A334710 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that three of them form a triangle of nonzero area and the extra point is on one of the edges of the triangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 6, 0, 0, 32, 48, 32, 0, 0, 100, 168, 168, 100, 0, 0, 240, 456, 532, 456, 240, 0, 0, 490, 990, 1312, 1312, 990, 490, 0, 0, 896, 1920, 2652, 3088, 2652, 1920, 896, 0, 0, 1512, 3360, 4972, 5964, 5964, 4972, 3360, 1512, 0, 0, 2400, 5520, 8420, 10816, 11340, 10816, 8420, 5520, 2400, 0

Offset: 1

N. J. A. Sloane, Jun 15 2020

Computed by Tom Duff, Jun 15 2020

			The initial rows of the array are:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 6, 32, 100, 240, 490, 896, 1512, 2400, 3630, 5280, ...
0, 6, 48, 168, 456, 990, 1920, 3360, 5520, 8550, 12720, 18216, ...
0, 32, 168, 532, 1312, 2652, 4972, 8420, 13452, 20480, 29980, 42288, ...
0, 100, 456, 1312, 3088, 5964, 10816, 17768, 27840, 41652, 60040, 83448, ...
0, 240, 990, 2652, 5964, 11340, 20142, 32436, 50004, 73704, 105282, 144936, ...
0, 490, 1920, 4972, 10816, 20142, 35264, 55916, 84960, 123690, 174976, 238512, ...
0, 896, 3360, 8420, 17768, 32436, 55916, 88088, 132708, 191588, 268972, 363876, ...
0, 1512, 5520, 13452, 27840, 50004, 84960, 132708, 198912, 285312, 397968, 534888, ...
0, 2400, 8550, 20480, 41652, 73704, 123690, 191588, 285312, 407744, 566046, 757008, ...
...
The initial antidiagonals are:
0
0, 0
0, 0, 0
0, 6, 6, 0
0, 32, 48, 32, 0
0, 100, 168, 168, 100, 0
0, 240, 456, 532, 456, 240, 0
0, 490, 990, 1312, 1312, 990, 490, 0
0, 896, 1920, 2652, 3088, 2652, 1920, 896, 0
0, 1512, 3360, 4972, 5964, 5964, 4972, 3360, 1512, 0
0, 2400, 5520, 8420, 10816, 11340, 10816, 8420, 5520, 2400, 0
0, 3630, 8550, 13452, 17768, 20142, 20142, 17768, 13452, 8550, 3630, 0
...

Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192

The main diagonal is A334713.
Triangles A334708, A334709, A334710, A334711 give the counts for the four possible arrangements of four points.
For three points there are just two possible arrangements: see A334704 and A334705.

A334713 Number of ways to select four points from an n X n grid so that three of them form a triangle of nonzero area and the extra point is on one of the edges of the triangle.

Original entry on oeis.org

0, 0, 48, 532, 3088, 11340, 35264, 88088, 198912, 407744, 783344, 1388196, 2392592, 3889292, 6124032, 9394536, 14058816, 20345400, 29034288, 40356156, 55487952, 75207892, 100445632, 131773104, 171900864, 221418336, 282240816, 356468556, 447353616, 555069252, 686394560, 840316880, 1023123264, 1238261952

Offset: 1

N. J. A. Sloane, Jun 15 2020

nonn

Computed by Tom Duff, Jun 15 2020 - Jun 22 2020

Tom Duff, Table of n, a(n) for n = 1..192
Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192

The main diagonal of A334710.

A334712 Number of ways to select four points from an n X n grid so that three of them form a triangle of nonzero area and the extra point is strictly inside the triangle.

Original entry on oeis.org

0, 0, 8, 240, 2100, 11604, 45184, 145648, 399656, 976552, 2172328, 4500816, 8736172, 16135996, 28493760, 48400296, 79500936, 126909720, 197253768, 299683032, 445664700, 650291092, 932742112, 1317391344, 1833532544, 2518622024, 3417812896, 4585946096, 6088659404, 8006284316, 10430982968, 13476148160

Offset: 1

N. J. A. Sloane, Jun 15 2020

nonn

Computed by Tom Duff, Jun 15 2020 - Jun 22 2020

Tom Duff, Table of n, a(n) for n = 1..192
Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192

The main diagonal of A334709.

cp's OEIS Frontend

A189413 Number of convex quadrilaterals on an n X n grid (or geoboard).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A334708 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four collinear points from an n X k grid.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A334709 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that three of them form a triangle of nonzero area and the extra point is strictly inside the triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A334710 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that three of them form a triangle of nonzero area and the extra point is on one of the edges of the triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A334713 Number of ways to select four points from an n X n grid so that three of them form a triangle of nonzero area and the extra point is on one of the edges of the triangle.

Views

Author

Keywords

Comments

Links

Crossrefs

A334712 Number of ways to select four points from an n X n grid so that three of them form a triangle of nonzero area and the extra point is strictly inside the triangle.

Views

Author

Keywords

Comments

Links

Crossrefs