A181944 Number of convex quadrilaterals, distinct up to congruence, on an n X n grid (or geoboard).

0, 1, 12, 89, 407, 1413, 3894, 9431, 20212, 39847, 73177, 127582, 211012, 337186, 519594, 777447, 1134269, 1620415, 2264873, 3114709, 4209184, 5609209, 7378581, 9594611, 12326333, 15688198, 19779188, 24721601, 30646522, 37727553, 46093734, 55983150, 67558997

Offset

1,3

Links

Table of n, a(n) for n=1..33.

Lucas A. Brown, Python program.

Eric Weisstein's World of Mathematics, Convex Polygon.

Eric Weisstein's World of Mathematics, Quadrilateral.

Example

a(1) = 0 because the 1 X 1 grid has no quadrilaterals.
a(2) = 1 because the 2 X 2 grid has one quadrilateral.
a(3) = 9 because the 3 X 3 grid has 12 congruence classes of quadrilaterals, out of 70 quadrilaterals 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 . | . . o |
| o . . | o . o | o . o | o . . |
| o . . | o . . | . o . | o o . |
+-------+-------+-------+-------+

Crossrefs

Cf. A189413.

Sequence in context: A009657 A012088 A083825 * A359311 A193285 A199558

Adjacent sequences: A181941 A181942 A181943 * A181945 A181946 A181947

Keyword

nonn

Author

Martin Renner, Apr 03 2012

Extensions

a(7)-a(33) from Lucas A. Brown, Feb 06 2024

Status

approved