A145581 -id:A145581 - cp's OEIS frontend

A145582 Number of isomorphism classes of rank one toric log del Pezzo surfaces with index L = n.

5, 7, 18, 13, 33, 26, 45, 27, 51, 51, 67, 53, 69, 74, 133, 48, 89, 81, 102, 110, 178, 105, 124, 109, 161, 119, 164, 135, 142, 187, 140, 105, 274, 159, 383, 169, 145, 166, 329, 221, 177, 266, 180, 230, 404, 189, 220, 213, 315, 264, 384, 233, 225, 260, 573, 298

Offset: 1

Jonathan Vos Post, Oct 14 2008

nonn

A145581(n) is the number of toric log del Pezzo surfaces with index L = n. Both are in the table for Theorem 1.2, p. 4 of Kasprzyk, Kreuzer and Nill.

Justus Springer, Table of n, a(n) for n = 1..5000
Andreas Bäuerle, Sharp volume and multiplicity bounds for Fano simplices, arXiv:2308.12719 [math.CO], 2023.
Andreas Bäuerle, Classification of Fano simplices
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. [see Proposition 7.1, p. 27]
Daniel Hättig, Jürgen Hausen, Justus Springer and Hendrik Süß, Log del Pezzo surfaces with torus action - a searchable database
Alexander M. Kasprzyk, Maximilian Kreuzer and Benjamin Nill, On the combinatorial classification of toric log del Pezzo surfaces, LMS J. Comput. Math. 13 (2010) 33-46; arXiv:0810.2207 [math.AG], 2008.
Alexander M. Kasprzyk and Benjamin Nill, Chapter 17 Fano polytopes, in: Strings, Gauge Fields, And The Geometry Behind, World Scientific, 2012. See p. 359.

Cf. A145581, A364712.

a(17) from Kasprzyk and Nill (2012) added by Andrey Zabolotskiy, Feb 17 2020
a(18)-a(200) from Haettig, Hafner, Hausen and Springer (2022) added by Justus Springer, Aug 04 2023
a(201)-a(1000) from Bäuerle's data, added by Andrey Zabolotskiy, Oct 01 2023
a(1001)-a(5000) computed using Bäuerle's algorithm, added by Justus Springer, Apr 15 2024

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.

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

Original entry on oeis.org

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.

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A145582 Number of isomorphism classes of rank one toric log del Pezzo surfaces with index L = n.

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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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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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