A070911 -id:A070911 - cp's OEIS frontend

A321693 Numerator of least value of the squared diameters of the enclosing circles of all strictly convex lattice n-gons with minimal area given by A070911.

2, 2, 50, 8, 10, 10, 1250, 29, 40, 52, 73, 73, 82, 82, 23290, 148, 202, 226, 317, 317, 365, 452, 500, 530

Offset: 3

Hugo Pfoertner, Nov 21 2018

Without the minimal area stipulation, the result differs for some n. (See n = 12 in the examples.) - Peter Munn, Nov 17 2022

			For n = 5, the polygon with minimal area A070911(5) = 5 and enclosing circle of least diameter is
  2         D
  |       +   +
  |     +       +
  |   +           +
  1 E               C
  | +             +
  | +          +
  | +        +
  0 A + + + B
    0 ----- 1 ----- 2 ---
.
The enclosing circle passes through points A (0,0), C (2,1) and D (1,2). Its diameter is sqrt(50/9). Therefore a(5) = 50 and A322029(5) = 9.
For n = 11, a strictly convex polygon ABCDEFGHIJKA with minimal area and enclosing circle of least diameter is
    0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6
  5                          J ++++++ I
  |                      +              +
  |                  +              .     +
  |             +                            +
  4         K                      .           H
  |       +                                      +
  |     +                        .                +
  |   +                                            +
  3 A                           .                   +
  | +    .                                           +
  | +           .             .                       +
  | +                 .                                +
  2 B                        O                          G
  |   +                            .                    +
  |     +                                 .             +
  |       +                                      .      +
  1         C                                           F
  |             +                                  +
  |                  +                        +
  |                      +                +
  0                          D ++++++ E
    0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6
.
The diameter d of the enclosing circle is determined by points A and F, with I also lying on this circle.  d^2 = 6^2 + 2^2 = 40. Therefore a(11) = 40 and A322029(11) = 1.
n = 12 is a case where the minimal area stipulation is significant. If we take the upper 6 edges in the n = 11 illustration above and rotate them about the enclosing circle's center to generate another 6 edges, we get a 12-gon with relevant squared diameter a(11) = 40 that meets all criteria except minimal area. This 12-gon's area is 26, and to meet the minimal area A070911(12)/2 = 24, the least squared diameter achievable is 52 (see illustration in the Pfoertner link). So a(12) = 52 and A322029(12) = 1. - _Peter Munn_, Nov 17 2022

Hugo Pfoertner, Illustrations of optimal polygons for n <= 26 (2018).

Cf. A070911, A192493, A192494, A322029 (corresponding denominators).

a(21)-a(26) from Hugo Pfoertner, Dec 03 2018

A322029 Denominator of least value of the squared diameters of the enclosing circles of all strictly convex lattice n-gons with minimal area given by A070911. Numerators are A321693.

Original entry on oeis.org

1, 1, 9, 1, 1, 1, 49, 1, 1, 1, 1, 1, 1, 1, 169, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 3

Hugo Pfoertner, Nov 24 2018

Cf. A070911, A192493, A192494, A321693.

a(21)-a(26) from Hugo Pfoertner, Dec 03 2018

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

Original entry on oeis.org

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

A063984 Minimal number of integer points in the Euclidean plane which are contained in the interior of any convex n-gon whose vertices have integer coordinates.

Original entry on oeis.org

0, 0, 1, 1, 4, 4, 7, 10, 17, 19, 27, 34, 45, 52, 68, 79, 98, 112, 135, 154, 183, 199, 237, 262, 300, 332, 378, 416, 469, 508, 573, 616, 688, 732, 818, 872, 959, 1020, 1120, 1202, 1305, 1391, 1504, 1598, 1724, 1815, 1961, 2064, 2220, 2332, 2497, 2625, 2785

Offset: 3

Pierre Bornsztein (pbornszt(AT)club-internet.fr), Sep 06 2001; May 20 2002

Consider convex lattice n-gons, that is, polygons whose n vertices are points on the integer lattice Z^2 and whose interior angles are strictly less than Pi. a(n) is the least possible number of lattice points in the interior of such an n-gon.
The result a(5) = 1 seems to be due to Ehrhart. Using Pick's formula, it is not difficult to prove that the determination of a(k) is equivalent to the determination of the minimal area of a convex k-gon whose vertices are lattice points.
Results before 2018 for odd n came from the following authors: a(3) (trivial), a(5) (Arkinstall), a(7) and a(9) (Rabinowitz), a(11) (Olszewska), a(13) (Simpson) and a(15) (Castryck). - Jamie Simpson, Oct 18 2022

			For example, every convex pentagon whose vertices are lattice points contains at least one lattice point and it is not difficult to construct such a pentagon with only one interior lattice point. Thus a(5) = 1.

I. Barany and N. Tokushige, The minimum area of convex lattice n-gons, Combinatorica, 24 (No. 2, 2004), 171-185.
Tian-Xin Cai, On the minimum area of convex lattice polygons, Taiwanese Journal of Mathematics, Vol. 1, No. 4 (1997).
W. Castryck, Moving Out the Edges of a Lattice Polygon, Discrete Comput. Geom., 47 (2012), pp. 496-518.
Code Golf StackExchange, The smallest area of a convex grid polygon, fastest-code challenge, started by Peter Kagey, Oct 22 2022, provides several programs.
C. J. Colbourn, R. J. Simpson, A note on bounds on the minimum area of convex lattice polygons, Bull. Austral. Math. Soc. 45 (1992) 237-240.
Steven R. Finch, Convex Lattice Polygons, December 18, 2003. [Cached copy, with permission of the author]
Hugo Pfoertner, Illustrations of optimal polygons for n <= 23, (2018).
S. Rabinowitz, O(n^3) bounds for the area of a convex lattice n-gon, Geombinatorics, vol. II, 4(1993), p. 85-88.
R. J. Simpson, Convex lattice polygons of minimum area, Bulletin of the Australian Math. Society, 42 (1990), pp. 353-367.

Cf. A070911, A089187, A321693, A322029, A322345.

a(n) = A070911(n)/2 - n/2 + 1. [Simpson]
See Barany & Tokushige for asymptotics.
a(n) = min(g: A322345(g) >= n). - Andrey Zabolotskiy, Apr 23 2023

Additional comments from Steven Finch, Dec 06 2003
More terms from Matthias Henze, Jul 27 2015
a(17)-a(23) from Hugo Pfoertner, Nov 27 2018
a(24)-a(25) from Hugo Pfoertner, Dec 04 2018
a(26)-a(55) from and definition clarified by Günter Rote, Sep 19 2023

A089187 a(n) is the minimal area of a convex lattice polygon with 2n sides.

Original entry on oeis.org

1, 3, 7, 14, 24, 40, 59, 87, 121, 164, 210, 274, 345, 430, 523, 632, 749, 890, 1039, 1222, 1412, 1620, 1838, 2088, 2357, 2651, 2953, 3278, 3612, 4020, 4439, 4902, 5387, 5898, 6418, 6974, 7557, 8182, 8835, 9512, 10218, 10984, 11759, 12635, 13525, 14448, 15399, 16415, 17473, 18570

Offset: 2

Jamie Simpson, Dec 07 2003

nonn

For polygons with an odd number of sides see A070911.

			The first entry is 1 because the convex lattice quadrilateral of minimal area is a unit square. The minimal area hexagon has area 3.

Günter Rote, Table of n, a(n) for n = 2..100
Charles J. Colbourn and R. J. Simpson, A note on bounds on the minimum area of convex lattice polygons, Bull. Austral. Math. Soc., 45[1992], 237-240.
Stanley Rabinowitz, Convex Lattice Polygons, Ph.D. Dissertation (Polytechnic University, Brooklyn, New York, 1986).
Günter Rote, a(n), together with coordinates of some smallest 2n-gon, for n=2..100, (2023).
Günter Rote, Python program for this sequence, and for A070911, (2023).
R. J. Simpson, Convex lattice polygons of minimum area, Bull. Austral. Math. Soc., 42[1990], 353-367.

The even-indexed subsequence of A070911. See also A063984.

a(22) onwards from Günter Rote, Sep 17 2023

A322106 Numerator of the least possible squared diameter of an enclosing circle of a strictly convex lattice n-gon.

Original entry on oeis.org

2, 2, 50, 8, 10, 10, 1250, 29, 40, 40, 2738, 72, 82, 82, 176900, 17810, 1709690, 178, 11300, 260, 290, 290, 568690, 416, 2418050, 488, 3479450, 629, 2674061, 730

Offset: 3

Hugo Pfoertner, Nov 26 2018

If the smallest possible enclosing circle is essentially determined by 3 vertices of the polygon, the squared diameter may be rational and thus A322107(n) > 1.
The first difference of the sequences A321693(n) / A322029(n) from a(n) / A322107(n) occurs for n = 12.
The ratio (A321693(n)/A322029(n)) / (a(n)/A322107(n)) will grow for larger n due to the tendency of the minimum area polygons to approach elliptical shapes with increasing aspect ratio, whereas the polygons leading to small enclosing circles will approach circular shape.
For n>=19, polygons with different areas may fit into the enclosing circle of minimal diameter. See examples in pdf at Pfoertner link.

			By n-gon a convex lattice n-gon is meant, area is understood omitting the factor 1/2. The following picture shows a comparison between the minimum area polygon and the polygon fitting in the smallest possible enclosing circle for n=12:
.
    0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6
  6                          H ##### Gxh +++++ g
  |                     #        +      #    *   +
  |                 #       +              #        +
  |             #       +                 *   #        +
  5         I       i                          F        f
  |       #       +                    *        #       +
  |     #       +                                #      +
  |   #       +                     *             #     +
  4 J       j                                      #    e
  | #     @+                     *                  #  +
  | #     +      @                                   #+
  | #    +              @     *                      +#
  3 K   +                     @                     +   E
  |  # +                   *         @             +    #
  |   #                                    @      +     #
  |  + #                *                        +@     #
  2 k   #                                      d        D
  | +    #           *                       +        #
  | +     #                                +        #
  | +      #       *                    +         #
  1 l       L                         c        C
  |   +       # *                +        #
  |     +       #           +        #
  |       +  *    #     +        #
  0         a ++++ Axb ##### B
    0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6
.
The 12-gon ABCDEFGHIJKLA with area 52 fits into a circle of squared diameter 40, e.g. determined by the distance D - J, indicated by @@@. No convex 12-gon with a smaller enclosing circle exists. Therefore a(n) = 40 and A322107(12) = 1.
For comparison, the 12-gon abcdefghijkla with minimal area A070911(12) = 48 requires a larger enclosing circle with squared diameter A321693(12)/A322029(12) = 52/1, e.g. determined by the distance a - g, indicated by ***.

See A063984.

Hugo Pfoertner, Illustration of convex n-gons fitting into smallest circle, (2018).
Hugo Pfoertner, Illustration of convex n-gons fitting into smallest circle, n = 27..32, (2018).

Cf. A063984, A070911, A321693, A322029, A322107 (corresponding denominators).

a(27)-a(32) from Hugo Pfoertner, Dec 19 2018

A322107 Denominator of the least possible squared diameter of an enclosing circle of a strictly convex lattice n-gon. Numerators are A322106.

Original entry on oeis.org

1, 1, 9, 1, 1, 1, 49, 1, 1, 1, 49, 1, 1, 1, 1369, 121, 9801, 1, 49, 1, 1, 1, 1521, 1, 5329, 1, 5929, 1, 3844, 1

Offset: 3

Hugo Pfoertner, Nov 26 2018

			See A322106.

Cf. A063984, A070911, A321693, A322029, A322106.

a(27)-a(32) from Hugo Pfoertner, Dec 19 2018

A357888 a(n) is the minimal squared length of the longest side of a strictly convex grid n-gon of smallest area.

Original entry on oeis.org

2, 1, 2, 2, 5, 2, 5, 5, 5, 5, 10, 5, 10, 5, 13, 10, 13, 10, 13, 13, 17, 13, 17, 13, 25, 17, 25, 17, 25, 13, 25, 17, 26, 17, 26, 17, 26, 17, 26, 25, 26, 25, 29, 29, 29, 34, 34, 34, 41, 37, 41, 37, 41, 34, 41, 41, 41, 41, 41, 41, 61, 41, 61, 41, 61, 41, 61, 41, 41

Offset: 3

Hugo Pfoertner, Nov 10 2022

nonn

It is conjectured that at least one polygon of smallest area exists with 4 sides of length 1 for n >= 8 and additionally 4 sides of squared length 2 for n >= 12.

Code Golf StackExchange, The smallest area of a convex grid polygon, fastest-code challenge, started by Peter Kagey, Oct 22 2022.
User AlephAlpha in Code Golf StackExchange, The smallest area of a convex grid polygon (n=7 .. 37), results of computations, Nov 2022.
Hugo Pfoertner, Illustrations of optimal polygons for n <= 26 (2018).
Günter Rote, Python program

Cf. A063984, A070911, A089187, A321693, A322029, A322345, A322348.

Python
```
# See Rote link.
```

a(29)-a(60) from Günter Rote, Sep 20 2023

Terms a(61) and beyond from Andrey Zabolotskiy, Sep 21 2023

cp's OEIS Frontend

A321693 Numerator of least value of the squared diameters of the enclosing circles of all strictly convex lattice n-gons with minimal area given by A070911.

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A322029 Denominator of least value of the squared diameters of the enclosing circles of all strictly convex lattice n-gons with minimal area given by A070911. Numerators are A321693.

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

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A063984 Minimal number of integer points in the Euclidean plane which are contained in the interior of any convex n-gon whose vertices have integer coordinates.

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A089187 a(n) is the minimal area of a convex lattice polygon with 2n sides.

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A322106 Numerator of the least possible squared diameter of an enclosing circle of a strictly convex lattice n-gon.

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A322107 Denominator of the least possible squared diameter of an enclosing circle of a strictly convex lattice n-gon. Numerators are A322106.

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Author

Keywords

Examples

Crossrefs

Extensions

A357888 a(n) is the minimal squared length of the longest side of a strictly convex grid n-gon of smallest area.

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