%I #20 Jun 22 2023 06:02:02
%S 1,1,2,4,11,30,89,250,776,2236,7147,20979,68448
%N Smallest number of triangulations of n points in the plane.
%C In a so-called double circle, half of the points are extremal, and for every edge of the convex hull, there is one interior point that is arbitrarily close to it. The double circles are conjectured to minimize the number of triangulations, and in this case the next terms would be 203748, 674949, 2031054, 6807382, 20662980, ... - _Manfred Scheucher_, Aug 22 2016
%D P. Brass, W. O. J. Moser, J. Pach, Research Problems in Discrete Geometry, Springer (2005).
%H O. Aichholzer, V. Alvarez, T. Hackl, A. Pilz, B. Speckmann and B. Vogtenhuber, <a href="https://doi.org/10.4230/LIPIcs.SoCG.2016.7">An Improved Lower Bound on the Minimum Number of Triangulations</a>, in Proceedings of the 32nd International Symposium on Computational Geometry (SoCG 2016), pages 7:1--7:16, LIPIcs, 2016.
%H O. Aichholzer, F. Hurtado, and M. Noy, <a href="http://www.ist.tugraz.at/files/publications/geometry/ahn-ntepp-01.ps.gz">On the Number of Triangulations Every Planar Point Set Must Have</a>, Proceedings of the 13th Annual Canadian Conference on Computational Geometry CCCG 2001, pages 13-16, Waterloo, Ontario, Canada, 2001. See also the <a href="http://www.ist.tugraz.at/aichholzer/research/rp/triangulations/counting/">Counting Triangulations - Olympics</a>
%H O. Aichholzer and H. Krasser, <a href="https://www.cccg.ca/proceedings/2001/hkrasser-17690.ps.gz">The point set order type data base: a collection of applications and results</a>, pp. 17-20 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
%H F. Santos and R. Seidel, <a href="https://doi.org/10.1016/S0097-3165(03)00002-5">A better upper bound on the number of triangulations of a planar point set</a>, Journal of Combinatorial Theory, Series A, 102(1):186-193, 2003.
%H EuroGIGA - CRP ComPoSe, <a href="https://www.eurogiga-compose.eu/posezo/double_circle/">Double Circle</a>
%F Conjecture: a(n) = sqrt(12)^(n-Theta(log n)). - _Manfred Scheucher_, Aug 22 2016
%Y Cf. A063545, A063666.
%K nonn,nice,hard,more
%O 3,3
%A _N. J. A. Sloane_, Aug 14 2001
%E a(11)-a(15) from _Manfred Scheucher_, Aug 22 2016