A322348 -id:A322348 - 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

A322349 Maximal lattice width of a convex lattice polygon of genus n, restricted to those polygons that are interior to another polygon.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Dec 04 2018

Sequence first differs from A322348 at n = 23.

See A322343.

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

Cf. A322343, A322348.

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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A322349 Maximal lattice width 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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