A298562 -id:A298562 - cp's OEIS frontend

A322345 Maximal number of vertices of a convex lattice polygon containing n lattice points in its interior.

4, 6, 6, 6, 8, 7, 8, 9, 8, 8, 10, 9, 9, 10, 10, 10, 10, 11, 10, 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, 16, 17, 17

Offset: 0

Hugo Pfoertner, Dec 04 2018

nonn

This is an inverse of A063984 in the following sense: A063984(k) = min {n : a(n)>=k}. Thus a(n) grows roughly like const*n^(1/3). - Günter Rote, Sep 19 2023

Günter Rote, Table of n, a(n) for n = 0..200
Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518, Column n_max in Table 1, p 512.
Wouter Castryck, Homepage. See the accompanying files for the above-referenced paper.
Günter Rote, Python program for this sequence and for A298562, (2023).
Günter Rote, Table of n, a(n) for n = 0..200 together with a corresponding a(n)-gon for each n, (2023).

Cf. A063984, A187015, A322343, A322346, A298562, A298755.

# See the Python program in the links section.

a(0) added by Andrey Zabolotskiy, Dec 29 2021
Name clarified by Günter Rote, Sep 18 2023
a(31) onwards from Günter Rote, Oct 01 2023

A298755 Quantitative (discrete) Helly numbers for the integer lattice Z^2.

Original entry on oeis.org

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

Bernardo González Merino, Jan 26 2018

nonn

a(n) = c(Z^2,n) is the smallest k>0 such that for every collection of convex sets C_1, ..., C_m having n points of Z^2 in common, there exists a subset of this collection of at most k elements such that they still contain exactly n points of Z^2 in common.

c(Z^2,n) = g(Z^2,n) = A298562(n) for n = 0, 1, ..., 200, but it is not known whether they agree for every n or not.

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.

Cf. A298562.

a(n) = max_{m=0..n} (A298562(m) + m - n). [Averkov et al.] - Andrey Zabolotskiy, Oct 02 2023

a(31) onwards from Andrey Zabolotskiy, Oct 02 2023

cp's OEIS Frontend

A322345 Maximal number of vertices of a convex lattice polygon containing n lattice points in its interior.

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