
COMMENTS

This means no three points on any line, not just lines in the X or Y directions.
A000755 gives the total number of solutions (as opposed to the number of equivalence classes).
It is conjectured that a(n)=0 for all sufficiently large n.
Flammenkamp's web site reports that at least one solution is known for all n <= 46 and n=48, 50, 52.
From R. K. Guy, Oct 22, 2004: (Start)
"I got the nothreeinline problem from Heilbronn over 50 years ago. See Section F4 in UPINT.
"In Canad. Math. Bull. 11 (1968) 527531, MR 39 #129, Guy & Kelly conjecture that, for large n, at most (c + eps)n points can be selected, where 3c^3 = 2pi^2 i.e. c ~ 1.85.
"As recently as last March, Gabor Ellmann pointed out an error in our heuristic reasoning, which, when corrected, gives 3c^2 = pi^2, or c ~ 1.813799." (End)


REFERENCES

M. A. Adena, D. A. Holton and P. A. Kelly, Some thoughts on the nothreeinline problem, pp. 617 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974.
D. B. Anderson, Journal of Combinatorial Theory Series A, V.27/1979 pp. 365  366.
D. Craggs and R. HughesJones, Journal of Combinatorial Theory Series A, V.20/1976 pp. 363  364
H. E. Dudeney, Amusements in Mathematics, Nelson, Edinburgh 1917, pp. 94, 222
A. Flammenkamp, Progress in the nothreeinline problem, J. Combinat. Theory A 60 (1992), 305311.
A. Flammenkamp, Progress in the nothreeinline problem. II. J. Combin. Theory Ser. A 81 (1998), no. 1, 108113.
M. Gardner, Scientific American V236 / March 1977, pp. 139  140
M. Gardner, Penrose Tiles to Trapdoor Ciphers. Freeman, NY, 1989, p. 69.
R. K. Guy, Unsolved combinatorial problems, pp. 121127 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.
R. K. Guy, Unsolved Problems Number Theory, Section F4.
R. K. Guy and P. A. Kelly, The NoThreeLine Problem. Research Paper 33, Department of Mathematics, Univ. of Calgary, Calgary, Alberta, 1968. Condensed version in Canad. Math. Bull. Vol. 11, pp. 527531, 1968.
R. R. Hall, T. H. Jackson, A. Sudberry and K. Wild, Journal of Combinatorial Theory Series A, V.18/1975 pp. 336  341
H. Harborth, P. Oertel and T. Prellberg, Discrete Math. V73/1988 pp. 8990
T. Klove, Journal of Combinatorial Theory Series A, V.24/1978 pp. 126  127
T. Klove, Journal of Combinatorial Theory Series A, V.26/1979 pp. 82  83
K. F. Roth, Journal London Math. Society V.26 / 1951, pp. 204
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).
