A000755 No-3-in-line problem on n X n grid: total number of ways of placing 2n points on n X n grid so no 3 are in a line. No symmetries are taken into account. (Formerly M1997 N0788)

0, 1, 2, 11, 32, 50, 132, 380, 368, 1135, 1120, 4348, 3622, 10568, 30634, 46304, 55576, 152210, 258176

Offset

1,3

Comments

This means no three on any line, not just lines in the X or Y directions.

References

M. A. Adena, D. A. Holton and P. A. Kelly, Some thoughts on the no-three-in-line problem, pp. 6-17 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974.

R. K. Guy, Unsolved combinatorial problems, pp. 121-127 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.

R. K. Guy and P. A. Kelly, The No-Three-Line Problem. Research Paper 33, Department of Mathematics, Univ. of Calgary, Calgary, Alberta, 1968. Condensed version in Canad. Math. Bull. Vol. 11, pp. 527-531, 1968.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..19.

Benjamin Chaffin, No-Three-In-Line Problem.

Achim Flammenkamp, Progress in the no-three-in-line problem

Achim Flammenkamp, Solutions of the no-three-in-line problem

Achim Flammenkamp, Solutions of the no-three-in-line problem

M. Gardner, R. L. Graham, M. Meierruth, R. Jacobson, Correspondence, 1976

R. K. Guy and P. A. Kelly, The No-Three-Line Problem, Research Paper 33, Department of Mathematics, Univ. of Calgary, Calgary, Alberta, 1968. [Annotated scanned copy]

R. K. Guy and P. A. Kelly, The No-Three-Line Problem, condensed version in Canad. Math. Bull. Vol. 11, pp. 527-531, 1968. [Annotated scanned copy]

R. K. Guy, P. A. Kelly, N. J. A. Sloane, Correspondence, 1968-1971

Mitchell Riley, C++/Cuda program (github)

Example

a(3) = 2:
X X o ... o X X
X o X ... X o X
o X X ... X X o

Crossrefs

Cf. A000769 (inequivalent solutions).

Sequence in context: A087933 A190259 A190261 * A183460 A033994 A023659

Adjacent sequences: A000752 A000753 A000754 * A000756 A000757 A000758

Keyword

nonn,nice,more

Author

N. J. A. Sloane

Extensions

More terms from the Achim Flammenkamp web site, May 24 2005

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

Minor edits from N. J. A. Sloane, May 25 2010

a(19) from Mitchell Riley, Feb 02 2026

Status

approved