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 A276096 a(n) is the least number of empty convex pentagons ("convex 5-holes") spanned by a set of n points in the Euclidean plane in general position (i.e., no three points on a line). 2
 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 3, 6, 9, 11 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,11 COMMENTS The value a(10) = 1 was determined by Harborth, who also constructed a set of 9 points without convex 5-holes. The values a(11) = 2 and a(13) = 3 were determined by Dehnhardt. Aichholzer found point sets showing that a(14) <= 6 and a(15) <= 9, and the exact values a(13) = 3, a(14) = 6, and a(15) = 9 were determined in the Bachelor's thesis of Scheucher, supervised by Aichholzer and Hackl. The value a(16) = 11 was determined using a ILP/SAT solver. For more information check out the link below with title "On 5-Holes". - Manfred Scheucher, Aug 18 2018 REFERENCES K. Dehnhardt, Leere konvexe Vielecke in ebenen Punktmengen, PhD thesis, TU Braunschweig, Germany, 1987, in German. LINKS O. Aichholzer, M. Balko, T. Hackl, J. Kynčl, I. Parada, M. Scheucher, P. Valtr, and B. Vogtenhuber, A superlinear lower bound on the number of 5-holes, arXiv:1703.05253 [math.CO], 2017. O. Aichholzer, R. Fabila-Monroy, T. Hackl, C. Huemer, A. Pilz, and B. Vogtenhuber, Lower bounds for the number of small convex k-holes, Computational Geometry: Theory and Applications, 47(5):605-613, 2014. EuroGIGA - CRP ComPoSe, A set of 13 points with 3 convex 5-holes EuroGIGA - CRP ComPoSe, A set of 14 points with 6 convex 5-holes EuroGIGA - CRP ComPoSe, A set of 15 points with 9 convex 5-holes EuroGIGA - CRP ComPoSe, A set of 16 points with 11 convex 5-holes H. Harborth, Konvexe Fünfecke in ebenen Punktmengen, Elemente der Mathematik, 33:116-118, 1978, in German. M. Scheucher, Counting Convex 5-Holes, Bachelor's thesis, Graz University of Technology, Austria, 2013, in German. M. Scheucher, On 5-Holes. FORMULA From Manfred Scheucher, Mar 22 2017: (Start) a(n) = Omega(n log^(4/5)(n)) and a(n) = O(n^2). Conjecture: a(n) = Theta(n^2). (End) CROSSREFS Cf. A063541 and A063542 for convex 3- and 4-holes, respectively. Cf. A006247 and A063666 for equivalence classes (w.r.t. orientation triples) of point sets in the plane. Sequence in context: A261090 A183560 A060840 * A074717 A218137 A320002 Adjacent sequences:  A276093 A276094 A276095 * A276097 A276098 A276099 KEYWORD nonn,more AUTHOR Manfred Scheucher, Aug 18 2016 EXTENSIONS a(16) from Manfred Scheucher, Mar 22 2017 STATUS approved

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Last modified July 23 17:32 EDT 2021. Contains 346259 sequences. (Running on oeis4.)