

A276096


a(n) is the least number of empty convex pentagons ("convex 5holes") 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
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OFFSET

1,11


COMMENTS

The value a(10) = 1 was determined by Harborth, who also constructed a set of 9 points without convex 5holes. 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 5Holes".  Manfred Scheucher, Aug 18 2018


REFERENCES

K. Dehnhardt, Leere konvexe Vielecke in ebenen Punktmengen, PhD thesis, TU Braunschweig, Germany, 1987, in German.


LINKS

Table of n, a(n) for n=1..16.
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 5holes, arXiv:1703.05253 [math.CO], 2017.
O. Aichholzer, R. FabilaMonroy, T. Hackl, C. Huemer, A. Pilz, and B. Vogtenhuber, Lower bounds for the number of small convex kholes, Computational Geometry: Theory and Applications, 47(5):605613, 2014.
EuroGIGA  CRP ComPoSe, A set of 13 points with 3 convex 5holes
EuroGIGA  CRP ComPoSe, A set of 14 points with 6 convex 5holes
EuroGIGA  CRP ComPoSe, A set of 15 points with 9 convex 5holes
EuroGIGA  CRP ComPoSe, A set of 16 points with 11 convex 5holes
H. Harborth, Konvexe Fünfecke in ebenen Punktmengen, Elemente der Mathematik, 33:116118, 1978, in German.
M. Scheucher, Counting Convex 5Holes, Bachelor's thesis, Graz University of Technology, Austria, 2013, in German.
M. Scheucher, On 5Holes.


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 4holes, 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



