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

3,1

Comments

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

Links

Table of n, a(n) for n=3..26.

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

Example

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

Crossrefs

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

Sequence in context: A316090 A345463 A322106 * A268536 A323872 A037292

Adjacent sequences: A321690 A321691 A321692 * A321694 A321695 A321696

Keyword

nonn,frac,hard

Author

Hugo Pfoertner, Nov 21 2018

Extensions

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

Status

approved