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%I #19 Jan 06 2025 05:51:49
%S 16,505,48032,1741603,154233886,2444400116
%N Number of rational polygons of denominator at most n having exactly one lattice point in their interior and primitive vertices, up to equivalence.
%C A rational polygon P of denominator d is said to have primitive vertices, if the lattice polygon d*P has primitive vertices.
%C 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.
%C 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.
%C 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.
%H Martin Bohnert and Justus Springer, <a href="https://arxiv.org/abs/2410.17244">Classifying rational polygons with small denominator and few interior lattice points</a>, arXiv:2410.17244 [math.CO], 2024.
%H Martin Bohnert and Justus Springer, <a href="https://doi.org/10.5281/zenodo.13839216">Rational polygons with exactly one interior lattice point</a> [Data set]. Zenodo.
%H Daniel Hättig, Jürgen Hausen, and Justus Springer, <a href="https://arxiv.org/abs/2302.03095">Classifying log del Pezzo surfaces with torus action</a>, arXiv:2302.03095 [math.AG], 2023.
%H Daniel Hättig, <a href="http://hdl.handle.net/10900/136648">Lattice Polygons and Surfaces with Torus Action</a>, Dissertation (2023).
%H Timo Hummel, <a href="http://hdl.handle.net/10900/112895">Automorphisms of rational projective K*-surfaces</a>, Dissertation (2021).
%H Justus Springer, <a href="https://github.com/justus-springer/RationalPolygons.jl">RationalPolygons.jl (Version 1.1.0) [Computer software]</a>, 2024.
%e For n = 1, there are 16 lattice polygons with exactly one interior lattice point, which are the 16 reflexive lattice polygons.
%Y Cf. A379887, A322343, A141682, A145581, A371917.
%K nonn,more,new
%O 1,1
%A _Justus Springer_, Jan 05 2025