A000769 -id:A000769 - cp's OEIS frontend

A280537 Maximum number of points that can be selected from an n X n X n grid so that no four of them are in a plane.

5, 8, 10, 13, 16, 18, 20

Offset: 2

Hugo Pfoertner, Jan 05 2017

Terms up to a(6) were found by exhaustive search. a(7) and a(8) are based on extensive numerical evidence.

Currently (January 2017) known lower bounds for the next terms are a(9)>=23, a(10)>=26, a(11)>=28, a(12)>=30, a(13)>=32, a(14)>=35, a(15)>=36, a(16)>=38, a(17)>=42.

Walter Möhres, Exhaustive Search for the 6x6x6 "No Four in Plane Problem". Private communication, September 2016.

Ed Pegg Jr, No-Four-In-Plane Problem, Wolfram Demonstrations Project.
Ed Pegg, No-Four-In-Plane, can 11 points be picked from a 4 X 4 X 4 grid?. Question in Mathematics Stack Exchange, a(4) and a(5) provided in answers.
Torsten Sillke, no 4 on a plane (3*3*3 puzzle), discussion in newsgroup rec.puzzles, Nov 27, 1992.
Al Zimmermann's Programming Contests, Non-Coplanar Points, March - June 2016.

Cf. A000769, A272651, A280538.

A037185 Number of ways of placing 2n points on n X n grid so no 3 are in a line (solutions with no symmetry).

Original entry on oeis.org

0, 0, 0, 0, 3, 4, 11, 40, 41, 132, 122, 524, 407, 1284, 3681, 5683, 6800, 18853

Offset: 1

N. J. A. Sloane

Benjamin Chaffin, No-Three-In-Line Problem.
A. Flammenkamp, Solutions of the no-three-in-line problem
A. Flammenkamp, Solutions of the no-three-in-line problem
A. Flammenkamp, Progress in the no-three-in-line problem, J. Combinat. Theory A 60 (1992), 305-311.
A. Flammenkamp, Progress in the no-three-in-line problem. II, J. Combin. Theory Ser. A 81 (1998), no. 1, 108-113.

Cf. A000769, A000755.

a(17) and a(18) from Benjamin Chaffin, Apr 05 2006

A037186 Number of ways of placing 2n points on n X n grid so no 3 are in a line (solutions with symmetry about one main diagonal).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 1, 5, 3, 3, 6, 3, 9, 5, 13, 14, 12, 14, 16, 17, 13, 18, 34, 43, 55

Offset: 1

N. J. A. Sloane

A. Flammenkamp, Solutions of the no-three-in-line problem
A. Flammenkamp, Progress in the no-three-in-line problem, J. Combinat. Theory A 60 (1992), 305-311.
A. Flammenkamp, Progress in the no-three-in-line problem. II, J. Combin. Theory Ser. A 81 (1998), no. 1, 108-113.

Cf. A000769.

A037187 Number of ways of placing 2n points on n X n grid so no 3 are in a line (solutions with 180 deg rotational symmetry).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 10, 7, 7, 13, 30, 33, 82, 61, 283, 189, 282, 328, 594, 675, 2413, 1248, 3968, 2852, 8983

Offset: 1

N. J. A. Sloane

A. Flammenkamp, Solutions of the no-three-in-line problem
A. Flammenkamp, Progress in the no-three-in-line problem, J. Combinat. Theory A 60 (1992), 305-311.
A. Flammenkamp, Progress in the no-three-in-line problem. II, J. Combin. Theory Ser. A 81 (1998), no. 1, 108-113.

Cf. A000769.

There are published versions of this sequence which incorrectly give 693 instead of 675.

A037188 Number of ways of placing 2n points on n X n grid so no 3 are in a line (solutions with symmetry about both main diagonals).

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 0, 0, 0, 1, 0, 2, 1, 3, 1, 1, 0, 2, 0, 2, 0, 1, 1, 2, 2, 0, 0, 0, 1, 1, 2, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

N. J. A. Sloane

A. Flammenkamp, Solutions of the no-three-in-line problem
A. Flammenkamp, Solutions of the no-three-in-line problem
A. Flammenkamp, Progress in the no-three-in-line problem, J. Combinat. Theory A 60 (1992), 305-311.
A. Flammenkamp, Progress in the no-three-in-line problem. II, J. Combin. Theory Ser. A 81 (1998), no. 1, 108-113.

Cf. A000769.

More terms from Flammenkamp web site, May 24 2005

A379299 a(n) is the maximum number k such that every permutation of the integers mod n admits at least k collinear triples.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 5, 2, 5, 0, 6, 9, 6, 4, 8

Offset: 1

Joshua Cooper, Dec 20 2024

Three points (x_i,y_i), i=1,2,3, are collinear if x_1*(y_2-y_3) + x_2*(y_3-y_1) + x_3*(y_1-y_2) == 0 (mod n).

Exhaustive search in SageMath obtained the reported values from Cooper and Solymosi 2004, where the authors show that (n-1)/4 <= a(n) <= (n-1)/2 for every odd prime n. In Li 2008, the author shows that a(n) = (n-1)/2 for every odd prime n.

			a(5)=2 because the permutation (in one-line notation) 0,1,3,2,4 admits two collinear triples mod 5: {(0,0),(1,1),(4,4)} is on the line y=x and {(0,0),(3,2),(2,3)} is on the line y=4*x; and all other permutations admit at least 2 collinear triples.

Joshua Cooper and Jack Hyatt, Permutations minimizing the number of collinear triples, arXiv:2501.02331 [math.CO], 2025. See p. 7.
Joshua N. Cooper and József Solymosi, Collinear points in permutations, Ann. Comb. 9 (2005), no. 2, 169-175; preprint, arXiv:math/0408396 [math.CO], 2004.
Liangpan Li, Collinear triples in permutations, Innov. Incidence Geom. 8 (2008), 171--173; arXiv preprint, arXiv:0805.0410 [math.CO], 2008.

Cf. A272651, A000769, A000755, A000938.

a(n) = (n-1)/2 for odd primes n.

cp's OEIS Frontend

A280537 Maximum number of points that can be selected from an n X n X n grid so that no four of them are in a plane.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A037185 Number of ways of placing 2n points on n X n grid so no 3 are in a line (solutions with no symmetry).

Views

Author

Keywords

Links

Crossrefs

Extensions

A037186 Number of ways of placing 2n points on n X n grid so no 3 are in a line (solutions with symmetry about one main diagonal).

Views

Author

Keywords

Links

Crossrefs

A037187 Number of ways of placing 2n points on n X n grid so no 3 are in a line (solutions with 180 deg rotational symmetry).

Views

Author

Keywords

Links

Crossrefs

Extensions

A037188 Number of ways of placing 2n points on n X n grid so no 3 are in a line (solutions with symmetry about both main diagonals).

Views

Author

Keywords

Links

Crossrefs

Extensions

A379299 a(n) is the maximum number k such that every permutation of the integers mod n admits at least k collinear triples.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula