A322344 -id:A322344 - cp's OEIS frontend

A322343 Number of equivalence classes of convex lattice polygons of genus n.

16, 45, 120, 211, 403, 714, 1023, 1830, 2700, 3659, 6125, 8101, 11027, 17280, 21499, 28689, 43012, 52736, 68557, 97733, 117776, 152344, 209409, 248983, 319957, 420714, 497676, 641229, 813814, 957001, 1214030, 1525951, 1774058, 2228111, 2747973, 3184761

Offset: 1

Hugo Pfoertner, Dec 04 2018

nonn

			a(1) = 16 because there are 16 equivalence classes of lattice polygons having exactly 1 interior lattice point. See Pfoertner link.

Justus Springer, Table of n, a(n) for n = 1..60
Wouter Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), p. 496-518, Column N in Table 1, p 512.
R. J. Koelman, The number of moduli families of curves on toric surfaces, Dissertation (1991), Chapter 4.2.
Hugo Pfoertner, Illustration of polygons of genus 1 representing the 16 equivalence classes, (2018).
B. Poonen and F. Rodriguez-Villegas, Lattice polygons and the number 12, Am. Math. Mon. 107 (2000), no. 3, 238-250 (2000).
Justus Springer, RationalPolygons.jl (Version 1.0.0) [Computer software], 2024.

Cf. A063984, A070911, A322344, A322345, A322346, A322347, A322348, A322349, A322350.

a(31) onwards from Justus Springer, Oct 25 2024

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.

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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A322343 Number of equivalence classes of convex lattice polygons of genus n.

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

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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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