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

%I A366409 #40 May 07 2025 09:40:48
%S A366409 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,
%T A366409 30,22,39,34,34,27,46,42,41,35,60,53,56,41,63,61,62,61,91,66,72,78,
%U A366409 111,87,86,83,135,123,111,97,142,135,156,146,176,148,186,194,206,169,200,242,313
%N A366409 Number of smooth convex lattice polygons with area n/2.
%C A366409 A lattice polygon is a polygon whose vertices have integer coordinates. (They belong to the integer lattice or grid Z x Z).
%C A366409 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.
%H A366409 Günter Rote, <a href="/A366409/b366409.txt">Table of n, a(n) for n = 1..300</a> (first 50 terms from Balletti (2021), Table 2 on p. 1114).
%H A366409 Gabriele Balletti, <a href="https://doi.org/10.1007/s00454-020-00187-y">Enumeration of lattice polytopes by their volume</a>, Discrete Comput. Geom., 65 (2021), 1087-1122.
%H A366409 Gabriele Balletti, <a href="https://github.com/gabrieleballetti/small-lattice-polytopes/blob/master/data/smooth/2_polytopes.txt">Dataset of "small" lattice polytopes</a> (2018).
%H A366409 T. Bogart, C. Haase, M. Hering, B. Lorenz, B. Nill, A. Paffenholz, G. Rote, F. Santos, and H. Schenck, <a href="https://doi.org/10.1007/s11856-015-1175-7">Finitely many smooth d-polytopes with n lattice points</a>, Israel Journal of Mathematics 207 (2015), 301-329; and <a href="https://arxiv.org/abs/1010.3887">arXiv version</a>, arXiv:1010.3887 [math.AG], 2010-2013.
%H A366409 Günter Rote, <a href="/A366409/a366409.py.txt">Python program</a> to count convex lattice polygons up to a given area (2023).
%H A366409 Günter Rote, <a href="/A366409/a366409.txt">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</a>.
%e A366409 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.)
%e A366409      V--V--+--+--+
%e A366409      |  |  |  |  |
%e A366409      V--I--B--+--+
%e A366409      |  |  |  |  |
%e A366409      +--V--I--B--+
%e A366409      |  |  |  |  |
%e A366409      +--+--+--V--V
%e A366409 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.
%e A366409 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.
%o A366409 (Python) # See the links section.
%Y A366409 Cf. A187015 for lattice polygons without the smoothness restriction. Cf. A127709.
%K A366409 nonn
%O A366409 1,4
%A A366409 _Günter Rote_, Oct 09 2023