A181944 -id:A181944 - 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

A181945 Number of trapezoids, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 9, 43, 141, 343, 766, 1415, 2517, 4129, 6545, 9505, 14230, 19444, 26733, 36208, 48029, 60675, 78729, 96866, 122433, 151288, 184072, 217998, 266775, 315096, 371138, 435153, 512549, 585240, 688470, 779196, 895058, 1019697, 1153081, 1305629, 1494185, 1656287

Offset: 1

Martin Renner, Apr 03 2012

nonn

			a(1) = 0 because the 1 X 1 grid has no trapezoids.
a(2) = 1 because the 2 X 2 grid has one trapezoid.
a(3) = 9 because the 3 X 3 grid has 9 congruence classes of trapezoids, out of 50 trapezoids total:
+-------+-------+-------+
| . . . | . o . | . . . |
| o o . | o . . | o . o |
| o o . | o o . | o . o |
+-------+-------+-------+
| . . o | o . o | . o . |
| o . . | . . . | o o . |
| o . o | o . o | o . . |
+-------+-------+-------+
| . o o | . . o | . o . |
| o . . | o . o | o . o |
| o . . | o . . | . o . |
+-------+-------+-------+

Lucas A. Brown, Python program.
Eric Weisstein's World of Mathematics, Trapezoid.

Cf. A181944, A189415.

a(7)-a(38) from Lucas A. Brown, Feb 05 2024

A181946 Number of kites, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 4, 11, 25, 45, 81, 121, 188, 261, 368, 469, 641, 785, 1000, 1220, 1520, 1767, 2161, 2471, 2961, 3396, 3946, 4403, 5164, 5744, 6517, 7227, 8201, 8936, 10122, 10963, 12240, 13312, 14649, 15839, 17607, 18813, 20482, 21983, 24111, 25589, 27920, 29550, 31979

Offset: 1

Martin Renner, Apr 03 2012

nonn

Only convex kites are counted, not concave kites (sometimes called darts or arrowheads).

			a(1) = 0 because the 1 X 1 grid has no kites.
a(2) = 1 because the 2 X 2 grid has one kite.
a(3) = 4 because the 3 X 3 grid has 4 congruence classes of kites, out of 10 kites total:
+-------+-------+-------+-------+
| . . . | o . o | . . o | . o . |
| o o . | . . . | o . . | o . o |
| o o . | o . o | o o . | . o . |
+-------+-------+-------+-------+

Lucas A. Brown, Python program.
Eric Weisstein's World of Mathematics, Kite.

Cf. A181944, A189417.

a(7)-a(45) from Lucas A. Brown, Feb 08 2024

A181947 Number of rhombi, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 3, 6, 11, 16, 24, 31, 43, 53, 67, 78, 99, 112, 132, 151, 179, 196, 226, 245, 282, 309, 341, 364, 416, 445, 483, 517, 570, 599, 659, 690, 754, 797, 847, 894, 975, 1012, 1068, 1119, 1211, 1252, 1338, 1381, 1466, 1536, 1604, 1651, 1775, 1833, 1923, 1990, 2091

Offset: 1

Martin Renner, Apr 03 2012

nonn

			a(1) = 0 because the 1 X 1 grid has no rhombi.
a(2) = 1 because the 2 X 2 grid has one rhombus.
a(3) = 3 because the 3 X 3 grid has 3 congruence classes of rhombi (all of which are squares) out of 6 rhombi total.
a(3) = 6 because the 4 X 4 grid has 6 congruence classes of rhombi, out of 22 rhombi total:
+---------+---------+---------+
| . . . . | . . . . | . . . . |
| . . . . | o . o . | . o . . |
| o o . . | . . . . | o . o . |
| o o . . | o . o . | . o . . |
+---------+---------+---------+
| o . . o | . . . o | . o . . |
| . . . . | . o . . | . . . o |
| . . . . | . . o . | o . . . |
| o . . o | o . . . | . . o . |
+---------+---------+---------+

Lucas A. Brown, Python program.
Eric Weisstein's World of Mathematics, Rhombus.

Cf. A181944, A189418.

a(7)-a(53) from Lucas A. Brown, Feb 08 2024

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

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A181945 Number of trapezoids, distinct up to congruence, on an n X n grid (or geoboard).

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A181946 Number of kites, distinct up to congruence, on an n X n grid (or geoboard).

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A181947 Number of rhombi, distinct up to congruence, on an n X n grid (or geoboard).

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