A187015 -id:A187015 - cp's OEIS frontend

A350408 Erroneous version of A187015.

1, 2, 3, 7, 6, 13, 11, 27

Offset: 1

dead

Sebastián Franco, Yang-Hui He, Chuang Sun and Yan Xiao, A comprehensive survey of brane tilings, Int. J. Mod. Phys. A, 32 (2017), 1750142, arXiv:1702.03958 [hep-th].
Heling Liu and Chuanming Zong, On the classification of convex lattice polytopes, Adv. Geom., 11 (2011), 711-729, arXiv:1103.0103 [math.MG]. See the table on p. 8.
Yan Xiao, Quivers, Tilings and Branes, City, University of London, 2018. See Tables 3.2-3.7.

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

A371917 Number of inequivalent convex lattice polygons containing n lattice points (including points on the boundary).

Original entry on oeis.org

1, 3, 6, 13, 21, 41, 67, 111, 175, 286, 419, 643, 938, 1370, 1939, 2779, 3819, 5293, 7191, 9752, 12991, 17321, 22641, 29687, 38533, 49796, 63621, 81300, 102807, 129787, 162833, 203642, 252898, 313666, 386601, 475540, 582216, 710688, 863552, 1048176

Offset: 3

Justus Springer, Apr 12 2024

nonn

A322343 counts the polygons by their number of interior lattice points, excluding points on the boundary.

			For n = 3, the only polygon is the standard triangle with vertices (0,0), (1,0) and (0,1).
For n = 4, a(4) = 3 and the three polygons have vertex sets {(1,0),(0,1),(-1,-1)}, {(0,0),(2,0),(0,1)} and {(0,0),(1,0),(0,1),(1,1)}.

Justus Springer, Table of n, a(n) for n = 3..112
Martin Bohnert and Justus Springer, Classifying rational polygons with small denominator and few interior lattice points, arXiv:2410.17244 [math.CO], 2024. See p. 20.
R. J. Koelman, The number of moduli families of curves on toric surfaces, Dissertation (1991), Chapter 4.4.
Justus Springer, RationalPolygons.jl (Version 1.0.0) [Computer software], 2024.
Justus Springer and M. Bohnert, Lattice polygons with at most 70 lattice points (1.0.0) [Data set], 2024.

Cf. A187015, A322343, A322344.

A366409 Number of smooth convex lattice polygons with area n/2.

Original entry on oeis.org

1, 1, 1, 3, 2, 4, 4, 6, 5, 7, 7, 9, 7, 12, 12, 15, 9, 15, 16, 18, 13, 23, 21, 24, 19, 26, 25, 30, 22, 39, 34, 34, 27, 46, 42, 41, 35, 60, 53, 56, 41, 63, 61, 62, 61, 91, 66, 72, 78, 111, 87, 86, 83, 135, 123, 111, 97, 142, 135, 156, 146, 176, 148, 186, 194, 206, 169, 200, 242, 313

Offset: 1

Günter Rote, Oct 09 2023

nonn

A lattice polygon is a polygon whose vertices have integer coordinates. (They belong to the integer lattice or grid Z x Z).

A convex lattice polygon is smooth if, for every vertex V, the adjacent lattice points on the two incident edges (which are not necessarily vertices) form together with V a triangle of area 1/2.

			Here is a smooth lattice polygon with k=6 vertices (V), 2 lattice points on edges (B), 2 interior lattice points (I), and area 5, shown as part of the grid: (The edges of the polygon are not drawn.)
     V--V--+--+--+
     |  |  |  |  |
     V--I--B--+--+
     |  |  |  |  |
     +--V--I--B--+
     |  |  |  |  |
     +--+--+--V--V
See Bogart et al., Theorem 32, and Appendix, p. 325, for a list of all 41 (convex) smooth lattice polygons with at most 12 lattice points, with figures.
The dataset of Balletti gives the complete set of 1530 polygons up to area 25. Beware that the vertices are not always listed in sorted (clockwise or counterclockwise) order around the polygon boundary.

Günter Rote, Table of n, a(n) for n = 1..300 (first 50 terms from Balletti (2021), Table 2 on p. 1114).
Gabriele Balletti, Enumeration of lattice polytopes by their volume, Discrete Comput. Geom., 65 (2021), 1087-1122.
Gabriele Balletti, Dataset of "small" lattice polytopes (2018).
T. Bogart, C. Haase, M. Hering, B. Lorenz, B. Nill, A. Paffenholz, G. Rote, F. Santos, and H. Schenck, Finitely many smooth d-polytopes with n lattice points, Israel Journal of Mathematics 207 (2015), 301-329; and arXiv version, arXiv:1010.3887 [math.AG], 2010-2013.
Günter Rote, Python program to count convex lattice polygons up to a given area (2023).
Günter Rote, Number of smooth lattice polygons of area at most 150, classified by the number k of vertices, the number B of lattice points on edges, and the number I of interior lattice points.

Cf. A187015 for lattice polygons without the smoothness restriction. Cf. A127709.

Python
```
# See the links section.
```

A374975 Number of equivalence classes of lattice polygons contained in a square of side length n but not in a square of side length n-1.

Original entry on oeis.org

2, 15, 131, 1369, 13842, 129185, 1104895, 8750964, 64714465, 450686225, 2976189422

Offset: 1

Justus Springer, Jul 26 2024

			For n = 1, only the square itself and the standard triangle are contained in the square, so a(1) = 2.

Gavin Brown and Alexander M. Kasprzyk, Small polygons and toric codes, J. Symbolic Comput. 51 (2013), pp. 55-62.
Justus Springer, RationalPolygons.jl (Version 1.0.0) [Computer software], 2024.
Justus Springer and Martin Bohnert, Classifying rational polygons with small denominator and few interior lattice points, arxiv:2410.17244 [math.CO], 2024.
Justus Springer and Martin Bohnert, Lattice subpolygons of a square with given sidelength (1.0.0) [Data set], 2024.

Cf. A187015, A322343, A371917.

A377245 Number of equivalence classes of convex lattice polygons containing n lattice points, restricting the count to those polygons that are interior to another polygon.

Original entry on oeis.org

1, 3, 4, 5, 7, 11, 16, 21, 25, 37, 46, 60, 69, 95, 110, 146, 179, 218, 258, 328, 378, 480, 557, 680, 792, 965, 1090, 1320, 1549, 1814, 2091, 2487, 2839, 3360, 3809, 4406, 5062, 5893, 6594, 7642, 8705, 9955, 11254, 12852, 14395, 16556, 18588, 20894, 23535

Offset: 3

Justus Springer, Oct 21 2024

nonn

See Castryck article for an explanation how to check if a polygon is interior to another polygon by application of theorem 5 (Koelman 1991).

The polygons up to 112 lattice points can be downloaded from the zenodo dataset linked below.

Justus Springer, Table of n, a(n) for n = 3..112
Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518.
R. J. Koelman, The number of moduli families of curves on toric surfaces, Dissertation (1991), Chapter 4.4.
Justus Springer, RationalPolygons.jl (Version 1.0.0) [Computer software], 2024.
Justus Springer and Martin Bohnert, Lattice polygons with at most 70 lattice points (1.1.0) [Data set], 2024.

Cf. A371917, A322343, A322344, A187015.

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A350408 Erroneous version of A187015.

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

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A371917 Number of inequivalent convex lattice polygons containing n lattice points (including points on the boundary).

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A366409 Number of smooth convex lattice polygons with area n/2.

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A374975 Number of equivalence classes of lattice polygons contained in a square of side length n but not in a square of side length n-1.

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A377245 Number of equivalence classes of convex lattice polygons containing n lattice points, restricting the count to those polygons that are interior to another polygon.

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