A189412 -id:A189412 - 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

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

A189414 Number of quadrilaterals on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 94, 1758, 13698, 70539, 266320, 837060, 2260984, 5460657, 12046294, 24775158, 47847538, 87967579, 154764912, 262110940, 429480216, 684015621, 1061192670, 1609449754, 2390049498, 3483126407

Offset: 1

Martin Renner, Apr 21 2011

nonn

Eric Weisstein's World of Mathematics, Quadrilateral.

Cf. A175383, A189345, A189412, A189413.

a(n) = A189412(n) + A189413(n).

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

a(7) - a(22) corrected by Michal Forisek, Sep 06 2011

A334581 Number of ways to choose 3 points that form an equilateral triangle from the A000292(n) points in a regular tetrahedral grid of side length n.

Original entry on oeis.org

0, 0, 4, 24, 84, 224, 516, 1068, 2016, 3528, 5832, 9256, 14208, 21180, 30728, 43488, 60192, 81660, 108828, 142764, 184708, 236088, 298476, 373652, 463524, 570228, 696012, 843312, 1014720, 1213096, 1441512, 1703352, 2002196, 2341848, 2726400, 3160272, 3648180

Offset: 0

Peter Kagey, May 06 2020

nonn

a(n) >= 4 * A269747(n).
a(n) >= 4 * A000389(n+3) = A210569(n+2).
a(n) >= 4 * (n-1) + 4 * a(n-1) - 6 * a(n-2) + 4 * a(n-3) - a(n-4) for n >= 4.

Peter Kagey, Table of n, a(n) for n = 0..1000 (Computed via Anders Kaseorg's program in the Stack Exchange link.)
Code Golf Stack Exchange, Triangles in a tetrahedron

Cf. A000292, A000389, A210569.
Cf. A000332 (equilateral triangles in triangular grid), A269747 (regular tetrahedra in a tetrahedral grid), A102698 (equilateral triangles in cube), A103158 (regular tetrahedra in cube).
Cf. A077435, A085582, A187452, A189412-A189418, A271910.

A334881 Number of squares in 3-dimensional space whose four vertices have coordinates (x,y,z) in the set {1,...,n}.

Original entry on oeis.org

0, 0, 6, 54, 240, 810, 2274, 5304, 10752, 19992, 34854, 57774, 91200, 139338, 206394, 296832, 417120, 575556, 779238, 1037514, 1359792, 1760694, 2251362, 2845140, 3554976, 4404876, 5416278, 6605946, 7996896, 9621678, 11500962, 13667772, 16143552, 18973608, 22190406

Offset: 0

Peter Kagey, May 14 2020

nonn

a(n) >= 3*n*A002415(n).

			For n = 5, one such square has vertex set {(2,1,1), (5,4,1), (4,5,5), (1,2,5)}.

Zachary Kaplan, Table of n, a(n) for n = 0..100
Zachary Kaplan, Python program
Mathematics Stack Exchange user Olivier Massicot, How many squares in a three-dimensional n X n X n cartesian grid?

Cf. A002415 (squares in square grid), A098928 (cubes in cube grid).

Cf. A000332, A077435, A085582, A102698, A103158, A187452, A189412-A189418, A269747, A271910, A334581.

a(7)-a(12) from Pontus von Brömssen, May 15 2020
a(13)-a(20) from Peter Kagey, Jul 29 2020 via Mathematics Stack Exchange link
Terms a(21) and beyond from Zachary Kaplan, Sep 01 2020, via Mathematics Stack Exchange link

A173502 Number of concave kites (darts or arrowheads) on an n X n grid (or geoboard).

Original entry on oeis.org

0, 0, 8, 64, 292, 916, 2344, 5048, 10096, 18296, 31400, 50752, 79308, 118876, 173768, 246096, 342256, 465064, 622904, 818552, 1063756, 1361564, 1725432, 2159384, 2682944, 3298568, 4027776, 4873600

Offset: 1

Martin Renner, Apr 29 2011

nonn

			a(1) = 0, since there is only one point on a 1 X 1 grid.
a(2) = 0, since the four points on a 2 X 2 grid build a convex quadrilateral.
a(3) = 8, since there are two possible forms of darts on a 3 X 3 grid, each can appear in four rotations.

Nathaniel Johnston, C program for computing terms

Cf. A189412, A189417.

a(7)-a(28) from Nathaniel Johnston, Apr 29 2011

cp's OEIS Frontend

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

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

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A189414 Number of quadrilaterals on an n X n grid (or geoboard).

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A334581 Number of ways to choose 3 points that form an equilateral triangle from the A000292(n) points in a regular tetrahedral grid of side length n.

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A334881 Number of squares in 3-dimensional space whose four vertices have coordinates (x,y,z) in the set {1,...,n}.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A173502 Number of concave kites (darts or arrowheads) on an n X n grid (or geoboard).

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Examples

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Extensions