A322345 -id:A322345 - cp's OEIS frontend

A322343 Number of equivalence classes of convex lattice polygons of genus n.

16, 45, 120, 211, 403, 714, 1023, 1830, 2700, 3659, 6125, 8101, 11027, 17280, 21499, 28689, 43012, 52736, 68557, 97733, 117776, 152344, 209409, 248983, 319957, 420714, 497676, 641229, 813814, 957001, 1214030, 1525951, 1774058, 2228111, 2747973, 3184761

Offset: 1

Hugo Pfoertner, Dec 04 2018

nonn

			a(1) = 16 because there are 16 equivalence classes of lattice polygons having exactly 1 interior lattice point. See Pfoertner link.

Justus Springer, Table of n, a(n) for n = 1..60
Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518, Column N in Table 1, p 512.
R. J. Koelman, The number of moduli families of curves on toric surfaces, Dissertation (1991), Chapter 4.2.
Hugo Pfoertner, Illustration of polygons of genus 1 representing the 16 equivalence classes, (2018).
B. Poonen and F. Rodriguez-Villegas, Lattice polygons and the number 12, Am. Math. Mon. 107 (2000), no. 3, 238-250 (2000).
Justus Springer, RationalPolygons.jl (Version 1.0.0) [Computer software], 2024.

Cf. A063984, A070911, A322344, A322345, A322346, A322347, A322348, A322349, A322350.

a(31) onwards from Justus Springer, Oct 25 2024

A063984 Minimal number of integer points in the Euclidean plane which are contained in the interior of any convex n-gon whose vertices have integer coordinates.

Original entry on oeis.org

0, 0, 1, 1, 4, 4, 7, 10, 17, 19, 27, 34, 45, 52, 68, 79, 98, 112, 135, 154, 183, 199, 237, 262, 300, 332, 378, 416, 469, 508, 573, 616, 688, 732, 818, 872, 959, 1020, 1120, 1202, 1305, 1391, 1504, 1598, 1724, 1815, 1961, 2064, 2220, 2332, 2497, 2625, 2785

Offset: 3

Pierre Bornsztein (pbornszt(AT)club-internet.fr), Sep 06 2001; May 20 2002

Consider convex lattice n-gons, that is, polygons whose n vertices are points on the integer lattice Z^2 and whose interior angles are strictly less than Pi. a(n) is the least possible number of lattice points in the interior of such an n-gon.
The result a(5) = 1 seems to be due to Ehrhart. Using Pick's formula, it is not difficult to prove that the determination of a(k) is equivalent to the determination of the minimal area of a convex k-gon whose vertices are lattice points.
Results before 2018 for odd n came from the following authors: a(3) (trivial), a(5) (Arkinstall), a(7) and a(9) (Rabinowitz), a(11) (Olszewska), a(13) (Simpson) and a(15) (Castryck). - Jamie Simpson, Oct 18 2022

			For example, every convex pentagon whose vertices are lattice points contains at least one lattice point and it is not difficult to construct such a pentagon with only one interior lattice point. Thus a(5) = 1.

I. Barany and N. Tokushige, The minimum area of convex lattice n-gons, Combinatorica, 24 (No. 2, 2004), 171-185.
Tian-Xin Cai, On the minimum area of convex lattice polygons, Taiwanese Journal of Mathematics, Vol. 1, No. 4 (1997).
W. Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), pp. 496-518.
Code Golf StackExchange, The smallest area of a convex grid polygon, fastest-code challenge, started by Peter Kagey, Oct 22 2022, provides several programs.
C. J. Colbourn, R. J. Simpson, A note on bounds on the minimum area of convex lattice polygons, Bull. Austral. Math. Soc. 45 (1992) 237-240.
Steven R. Finch, Convex Lattice Polygons, December 18, 2003. [Cached copy, with permission of the author]
Hugo Pfoertner, Illustrations of optimal polygons for n <= 23, (2018).
S. Rabinowitz, O(n^3) bounds for the area of a convex lattice n-gon, Geombinatorics, vol. II, 4(1993), p. 85-88.
R. J. Simpson, Convex lattice polygons of minimum area, Bulletin of the Australian Math. Society, 42 (1990), pp. 353-367.

Cf. A070911, A089187, A321693, A322029, A322345.

a(n) = A070911(n)/2 - n/2 + 1. [Simpson]
See Barany & Tokushige for asymptotics.
a(n) = min(g: A322345(g) >= n). - Andrey Zabolotskiy, Apr 23 2023

Additional comments from Steven Finch, Dec 06 2003
More terms from Matthias Henze, Jul 27 2015
a(17)-a(23) from Hugo Pfoertner, Nov 27 2018
a(24)-a(25) from Hugo Pfoertner, Dec 04 2018
a(26)-a(55) from and definition clarified by Günter Rote, Sep 19 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

A322348 Maximal lattice width of a convex lattice polygon containing n lattice points in its interior ("of genus n").

Original entry on oeis.org

2, 3, 2, 4, 4, 4, 5, 4, 4, 5, 6, 5, 6, 6, 6, 7, 6, 6, 7, 8, 7, 8, 8, 8, 8, 8, 8, 8, 9, 8, 9

Offset: 0

Hugo Pfoertner, Dec 04 2018

Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518, Column lw_max in Table 1, p 512.

Cf. A322343, A322345, A322349.

a(0) added and name clarified by Andrey Zabolotskiy, Sep 21 2023

A322346 Maximal number of vertices of a convex lattice polygon of genus n, restricted to those polygons that are interior to another polygon.

Original entry on oeis.org

6, 6, 6, 8, 7, 8, 9, 8, 8, 10, 9, 9, 10, 10, 10, 10, 11, 10, 12, 11, 12, 11, 11, 12, 12, 12, 12, 12, 12, 12

Offset: 1

Hugo Pfoertner, Dec 04 2018

See A322343.

Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518, Column n_max(1) in Table 1, p 512.

Cf. A322343, A322345.

A357888 a(n) is the minimal squared length of the longest side of a strictly convex grid n-gon of smallest area.

Original entry on oeis.org

2, 1, 2, 2, 5, 2, 5, 5, 5, 5, 10, 5, 10, 5, 13, 10, 13, 10, 13, 13, 17, 13, 17, 13, 25, 17, 25, 17, 25, 13, 25, 17, 26, 17, 26, 17, 26, 17, 26, 25, 26, 25, 29, 29, 29, 34, 34, 34, 41, 37, 41, 37, 41, 34, 41, 41, 41, 41, 41, 41, 61, 41, 61, 41, 61, 41, 61, 41, 41

Offset: 3

Hugo Pfoertner, Nov 10 2022

nonn

It is conjectured that at least one polygon of smallest area exists with 4 sides of length 1 for n >= 8 and additionally 4 sides of squared length 2 for n >= 12.

Code Golf StackExchange, The smallest area of a convex grid polygon, fastest-code challenge, started by Peter Kagey, Oct 22 2022.
User AlephAlpha in Code Golf StackExchange, The smallest area of a convex grid polygon (n=7 .. 37), results of computations, Nov 2022.
Hugo Pfoertner, Illustrations of optimal polygons for n <= 26 (2018).
Günter Rote, Python program

Cf. A063984, A070911, A089187, A321693, A322029, A322345, A322348.

Python
```
# See Rote link.
```

a(29)-a(60) from Günter Rote, Sep 20 2023

Terms a(61) and beyond from Andrey Zabolotskiy, Sep 21 2023

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A322343 Number of equivalence classes of convex lattice polygons of genus n.

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A063984 Minimal number of integer points in the Euclidean plane which are contained in the interior of any convex n-gon whose vertices have integer coordinates.

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

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A322348 Maximal lattice width of a convex lattice polygon containing n lattice points in its interior ("of genus n").

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A322346 Maximal number of vertices of a convex lattice polygon of genus n, restricted to those polygons that are interior to another polygon.

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A357888 a(n) is the minimal squared length of the longest side of a strictly convex grid n-gon of smallest area.

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