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A379887 Number of rational polygons with denominator at most n having exactly one lattice point in their interior, up to equivalence.

16, 5145, 924042, 101267212, 8544548186

Offset: 1

Justus Springer, Jan 05 2025

A379894 counts the polygons with the extra condition that the vertices are primitive. Both are in Classification 5.6 of the article by Bohnert and Springer.

An algorithm to compute a(n) was given by Timo Hummel in his dissertation. His final number for n = 3 (given in Theorem 12.1) was however slightly off.

			For n = 1, there are 16 lattice polygons with exactly one interior lattice point, which are the 16 reflexive lattice polygons.

Martin Bohnert and Justus Springer, Classifying rational polygons with small denominator and few interior lattice points, arXiv:2410.17244 [math.CO], 2024.
Martin Bohnert and Justus Springer, Rational polygons with exactly one interior lattice point [Data set]. Zenodo.
Daniel Hättig, Jürgen Hausen, and Justus Springer, Classifying log del Pezzo surfaces with torus action, arXiv:2302.03095 [math.AG], 2023.
Daniel Hättig, Lattice Polygons and Surfaces with Torus Action, Dissertation (2023).
Timo Hummel, Automorphisms of rational projective K*-surfaces, Dissertation (2021).
Justus Springer, RationalPolygons.jl (Version 1.1.0) [Computer software], 2024.

Cf. A379894, A322343, A145581, A371917.

A379894 Number of rational polygons of denominator at most n having exactly one lattice point in their interior and primitive vertices, up to equivalence.

Original entry on oeis.org

16, 505, 48032, 1741603, 154233886, 2444400116

Offset: 1

Justus Springer, Jan 05 2025

A rational polygon P of denominator d is said to have primitive vertices, if the lattice polygon d*P has primitive vertices.
A379887 counts the polygons without the condition that the vertices are primitive. Both are in Classification 5.6 of the article by Bohnert and Springer.
a(n) is also the number of isomorphism classes of 1/n-log canonical toric del Pezzo surfaces, see the article by Hättig, Hausen, Hafner and Springer.
An algorithm to compute a(n) was given by Timo Hummel in his dissertation. His final number for n = 3 (given in Corollary 12.2) was however slightly off.

			For n = 1, there are 16 lattice polygons with exactly one interior lattice point, which are the 16 reflexive lattice polygons.

Martin Bohnert and Justus Springer, Classifying rational polygons with small denominator and few interior lattice points, arXiv:2410.17244 [math.CO], 2024.
Martin Bohnert and Justus Springer, Rational polygons with exactly one interior lattice point [Data set]. Zenodo.
Daniel Hättig, Jürgen Hausen, and Justus Springer, Classifying log del Pezzo surfaces with torus action, arXiv:2302.03095 [math.AG], 2023.
Daniel Hättig, Lattice Polygons and Surfaces with Torus Action, Dissertation (2023).
Timo Hummel, Automorphisms of rational projective K*-surfaces, Dissertation (2021).
Justus Springer, RationalPolygons.jl (Version 1.1.0) [Computer software], 2024.

Cf. A379887, A322343, A141682, A145581, 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.

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.

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.

A364712 Number of families of non-toric log del Pezzo surfaces of Picard number one with Gorenstein index = n that admit an effective action of a one-dimensional torus.

Original entry on oeis.org

13, 10, 36, 25, 80, 37, 100, 56, 109, 71, 176, 85, 158, 105, 200, 102, 226, 102, 241, 178, 253, 150, 312, 176, 269, 149, 336, 224, 395, 192, 309, 216, 381, 207, 592, 230, 336, 239, 497, 312, 481, 266, 405, 348, 526, 270, 549, 317, 497, 277, 570, 354, 532, 334

Offset: 1

Justus Springer, Aug 04 2023

nonn

This sequence appears in Proposition 7.1, p. 27 of Haettig, Hafner, Hausen and Springer.

Justus Springer, Table of n, a(n) for n = 1..200
Daniel Haettig, Beatrice Hafner, Juergen Hausen and Justus Springer, Del Pezzo surfaces of Picard number one admitting a torus action, arXiv:2207.14790 [math.AG], 2022.
Daniel Hättig, Jürgen Hausen, Justus Springer and Hendrik Süß, Log del Pezzo surfaces with torus action - a searchable database

Cf. A145582, A145581.

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User: Justus Springer

A379887 Number of rational polygons with denominator at most n having exactly one lattice point in their interior, up to equivalence.

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A379894 Number of rational polygons of denominator at most n having exactly one lattice point in their interior and primitive vertices, up to equivalence.

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

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A364712 Number of families of non-toric log del Pezzo surfaces of Picard number one with Gorenstein index = n that admit an effective action of a one-dimensional torus.

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