

A366409


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


2



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, 30, 22, 39, 34, 34, 27, 46, 42, 41, 35, 60, 53, 56, 41, 63, 61, 62, 61, 91, 66, 72, 78, 111, 87, 86, 83, 135, 123, 111, 97, 142, 135, 156, 146, 176, 148, 186, 194, 206, 169, 200, 242, 313
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,4


COMMENTS

A lattice polygon is a polygon whose vertices have integer coordinates. (They belong to the integer lattice or grid Z x Z).
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.


LINKS

T. Bogart, C. Haase, M. Hering, B. Lorenz, B. Nill, A. Paffenholz, G. Rote, F. Santos, and H. Schenck, Finitely many smooth dpolytopes with n lattice points, Israel Journal of Mathematics 207 (2015), 301329; and arXiv version, arXiv:1010.3887 [math.AG], 20102013.
Günter Rote, Python program to count convex lattice polygons up to a given area (2023).


EXAMPLE

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.)
VV+++
    
VIB++
    
+VIB+
    
+++VV
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.
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.


PROG

(Python) see the links section.


CROSSREFS

Cf. A187015 for lattice polygons without the smoothness restriction. Cf. A127709.


KEYWORD

nonn


AUTHOR



STATUS

approved



