A298755 -id:A298755 - 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

A298562 Quantitative (polygonal) 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 21 2018

nonn

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]

			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)].

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).

Cf. A298755, A322345.

Python
```
# See the Python program for A322345.
```

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Extensions