Search: id:A000769
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%I A000769 M3252 N1313
%S A000769 1,1,4,5,11,22,57,51,156,158,566,499,1366,3978,5900,7094,19204
%N A000769 No-3-in-line problem: number of ways of placing 2n points on n X n grid 
               so no 3 are in a line.
%C A000769 This means no three on any line, not just lines in the X or Y directions.
%C A000769 Comments from R. K. Guy, Oct 22, 2004: "I got the no-three-in-line problem 
               from Heilbronn over 50 years ago. See SectionF4 in UPINT.
%C A000769 "In Canad. Math. Bull. 11 (1968) 527-531, 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.
%C A000769 "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."
%D A000769 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.
%D A000769 D. B. Anderson, Journal of Combinatorial Theory Series A, V.27/1979 pp. 
               365 - 366
%D A000769 D. Craggs and R. Hughes-Jones, Journal of Combinatorial Theory Series 
               A, V.20/1976 pp. 363 - 364
%D A000769 H. E. Dudeney, Amusements in Mathematics, Nelson, Edinburgh 1917, pp. 
               94, 222
%D A000769 A. Flammenkamp, Progress in the no-three-in-line problem, J. Combinat. 
               Theory A 60 (1992), 305-311.
%D A000769 A. Flammenkamp, Progress in the no-three-in-line problem. II. J. Combin. 
               Theory Ser. A 81 (1998), no. 1, 108-113.
%D A000769 M. Gardner, Scientific American V236 / March 1977, pp. 139 - 140
%D A000769 M. Gardner, Penrose Tiles to Trapdoor Ciphers. Freeman, NY, 1989, p. 
               69.
%D A000769 R. K. Guy, Unsolved combinatorial problems, pp. 121-127 of D. J. A. Welsh, 
               editor, Combinatorial Mathematics and Its Applications. Academic 
               Press, NY, 1971.
%D A000769 R. K. Guy, Unsolved Problems Number Theory, Section F4.
%D A000769 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.
%D A000769 R. R. Hall, T. H. Jackson, A. Sudberry and K. Wild, Journal of Combinatorial 
               Theory Series A, V.18/1975 pp. 336 - 341
%D A000769 H. Harborth, P. Oertel and T. Prellberg, Discrete Math. V73/1988 pp. 
               89-90
%D A000769 T. Klove, Journal of Combinatorial Theory Series A, V.24/1978 pp. 126 
               - 127
%D A000769 T. Klove, Journal of Combinatorial Theory Series A, V.26/1979 pp. 82 
               - 83
%D A000769 K. F. Roth, Journal London Math. Society V.26 / 1951, pp. 204
%D A000769 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000769 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000769 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/no3in/
               readme.html">Progress in the no-three-in-line problem</a>
%H A000769 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/no3in/
               table.txt">Solutions of the no-three-in-line problem</a>
%H A000769 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PointLattice.html">Link to a section of The World of Mathematics.</
               a>
%H A000769 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               No-Three-in-a-Line-Problem.html">No-Three-in-a-Line-Problem</a>
%H A000769 Benjamin Chaffin, <a href="http://wso.williams.edu/~bchaffin/no_three_in_line/
               index.htm">No-Three-In-Line Problem</a>.
%Y A000769 Cf. A000755, A037185, A037186, A037187, A037188, A037189, A047840.
%Y A000769 Sequence in context: A077238 A000286 A036539 this_sequence A050831 A056799 
               A109503
%Y A000769 Adjacent sequences: A000766 A000767 A000768 this_sequence A000770 A000771 
               A000772
%K A000769 hard,nonn,nice,fini
%O A000769 2,3
%A A000769 N. J. A. Sloane (njas(AT)research.att.com).
%E A000769 It is known 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.
%E A000769 a(17) and a(18) from Benjamin Chaffin (chaffin(AT)gmail.com), Apr 05 
               2006

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