A298562 Quantitative (polygonal) Helly numbers for the integer lattice Z^2.

4, 6, 6, 6, 8, 7, 8, 9, 8, 8, 10, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 11, 11, 12, 12, 12, 13, 12, 12, 13, 13, 13, 13, 14, 14, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 14, 15, 15, 15, 15, 15, 16, 15, 16, 15, 16, 16, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 16, 17, 17, 17, 17, 17

Offset: 0

Author

Bernardo González Merino, Jan 21 2018

Keywords

• nonn

Comments

a(n) = g(Z^2,n) is the maximum integer k > 0 such that there exists a lattice polygon with k vertices containing exactly n+k lattice points (in its interior or on the boundary). [edited by Günter Rote, Oct 01 2023]

Examples

			a(18) = 11 (so this sequence differs from A322345), attained only by the following polygon (No. 3736 in the corresponding list in Castryck's file) with 11 vertices, 1 non-vertex boundary lattice point, and genus (number of internal lattice points) 17: [(-2, -1), (-1, -2), (1, -2), (3, -1), (4, 0), (4, 1), (3, 2), (1, 3), (0, 3), (-1, 2), (-2, 0)].
		

Links

• Günter Rote, Table of n, a(n) for n = 0..200
• G. Averkov, B. González Merino, I. Paschke, M. Schymura, and S. Weltge, Tight bounds on discrete quantitative Helly numbers, arXiv:1602.07839 [math.CO], 2016. See Fig. 3 p. 5.
• G. Averkov, B. González Merino, I. Paschke, M. Schymura, and S. Weltge, Tight bounds on discrete quantitative Helly numbers, Adv. in Appl. Math., 89 (2017), 76--101.
• Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518.
• Wouter Castryck, Homepage. See the accompanying files for the above-referenced paper.
• Günter Rote, Table of n, a(n) for n = 0..200 together with a corresponding a(n)-gon for each n, (2023).

Crossrefs

Cf. A298755, A322345.

Programs

Python

# See the Python program for A322345.

Extensions

a(31) onwards from Günter Rote, Oct 01 2023