%I A001403
%S A001403 0,0,0,0,0,0,1,1,3,10,31,229,2036,21399,245342,3004881,38904499,
%T A001403 530452205,7640941062
%N A001403 Number of combinatorial configurations of type (n_3).
%C A001403 A combinatorial configuration of type (n_3) consists of an (abstract) 
               set of n points together with a set of n triples of points, called 
               lines, such that each point belongs to 3 lines and each line contains 
               3 points.
%D A001403 A. Betten and D. Betten, Regular linear spaces, Beitraege zur Algebra 
               und Geometrie, 38 (1997), 111-124.
%D A001403 A. Betten and D. Betten, Tactical decompositions and some configurations 
               v_4, J. Geom. 66 (1999), 27-41.
%D A001403 A. Betten, G. Brinkmann and T. Pisanski, Counting symmetric configurations 
               v_3, Discrete Appl. Math., 99 (2000), 331-338.
%D A001403 M. Boben et al., Small triangle-free configurations of points and lines, 
               Discrete Comput. Geom., 35 (2006), 405-427.
%D A001403 Bokowski and Sturmfels, Comput. Synthetic Geom., Lect Notes Math. 1355, 
               p. 41.
%D A001403 CRC Handbook of Combinatorial Designs, 1996, p. 255.
%D A001403 H. Gropp, Configurations and their realization, Discr. Math. 174 (1997), 
               137-151.
%D A001403 D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination Chelsea, 
               NY, 1952, Ch. 3.
%D A001403 Pisanski, T.; Boben, M.; Marusic, D.; Orbanic, A.; and Graovac, A. The 
               10-cages and derived configurations. Discrete Math. 275 (2004), 265-276.
%D A001403 Pisanski, T. and Randic, M., Bridges between Geometry and Graph Theory, 
               in Geometry at Work: Papers in Applied Geometry (Ed. C. A. Gorini), 
               M.A.A., Washington, DC, pp. 174-194, 2000.
%D A001403 B. Polster, A Geometrical Picture Book, Springer, 1998, p. 28.
%D A001403 Sturmfels and White, Rational realizations..., in H. Crapo et al. editors, 
               Symbolic Computation in Geometry, IMA preprint, Univ Minn., 1988.
%D A001403 Sturmfels and White, All 11_3 and 12_3 configurations are rational, Aeq. 
               Math., 39 1990 254-260.
%D A001403 Von Sterneck, Die Config. 11_3, Monat. f. Math. Phys. 5 325-330 1894; 
               Die Config. 12_3, op. cit. 6 223-255 1895.
%H A001403 Jim Loy, <a href="http://www.jimloy.com/math/math.htm">Mathematics Page</
               a> (see Desargues's Theorem)
%H A001403 Jim Loy, <a href="a099999.gif">The configuration (10_3) arising from 
               Desargues's theorem</a>
%H A001403 Tomo Pisanski, <a href="http://www.ijp.si/Configurations2004/papers.html">
               Papers on configurations</a>
%H A001403 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Configuration.html">Link to a section of The World of Mathematics.</
               a>
%e A001403 Example: the Fano plane is the only (7_3) configuration. It contains 
               7 points 1,2,...7 and 7 triples, 124, 235, 346, 457, 561, 672, 713.
%e A001403 The unique (8_3) configuration consists of the triples 125, 148, 167, 
               236, 278, 347, 358, 456.
%e A001403 There are three configurations (9_3), one of which is the familiar configuration 
               arising from Desargues's theorem (see illustration).
%Y A001403 Cf. A023994, A099999 (geometrical configurations), A100001 (self-dual 
               configurations), A098702, A098804, A098822, A098841, A098851, A098852, 
               A098854.
%Y A001403 Sequence in context: A079522 A024426 A034016 this_sequence A072136 A080406 
               A036682
%Y A001403 Adjacent sequences: A001400 A001401 A001402 this_sequence A001404 A001405 
               A001406
%K A001403 nonn,nice,hard
%O A001403 1,9
%A A001403 N. J. A. Sloane (njas(AT)research.att.com), D.Glynn(AT)math.canterbury.ac.nz
%E A001403 Von Sterneck has 228 instead of 229. His error was corrected by Gropp. 
               The n=15 term was computed by Dieter and Anton Betten, Uni. of Kiel.
%E A001403 a(16)-a(18) from the Betten, Brinkmann and Pisanski article.
%E A001403 a(19) from the Pisanski et al. article.