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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.

%I A321693 #37 Nov 18 2022 09:23:28
%S A321693 2,2,50,8,10,10,1250,29,40,52,73,73,82,82,23290,148,202,226,317,317,
%T A321693 365,452,500,530
%N 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.
%C A321693 Without the minimal area stipulation, the result differs for some n. (See n = 12 in the examples.) - _Peter Munn_, Nov 17 2022
%H A321693 Hugo Pfoertner, <a href="/A321693/a321693_1.pdf">Illustrations of optimal polygons for n <= 26</a> (2018).
%e A321693 For n = 5, the polygon with minimal area A070911(5) = 5 and enclosing circle of least diameter is
%e A321693   2         D
%e A321693   |       +   +
%e A321693   |     +       +
%e A321693   |   +           +
%e A321693   1 E               C
%e A321693   | +             +
%e A321693   | +          +
%e A321693   | +        +
%e A321693   0 A + + + B
%e A321693     0 ----- 1 ----- 2 ---
%e A321693 .
%e A321693 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.
%e A321693 For n = 11, a strictly convex polygon ABCDEFGHIJKA with minimal area and enclosing circle of least diameter is
%e A321693     0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6
%e A321693   5                          J ++++++ I
%e A321693   |                      +              +
%e A321693   |                  +              .     +
%e A321693   |             +                            +
%e A321693   4         K                      .           H
%e A321693   |       +                                      +
%e A321693   |     +                        .                +
%e A321693   |   +                                            +
%e A321693   3 A                           .                   +
%e A321693   | +    .                                           +
%e A321693   | +           .             .                       +
%e A321693   | +                 .                                +
%e A321693   2 B                        O                          G
%e A321693   |   +                            .                    +
%e A321693   |     +                                 .             +
%e A321693   |       +                                      .      +
%e A321693   1         C                                           F
%e A321693   |             +                                  +
%e A321693   |                  +                        +
%e A321693   |                      +                +
%e A321693   0                          D ++++++ E
%e A321693     0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6
%e A321693 .
%e A321693 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.
%e A321693 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
%Y A321693 Cf. A070911, A192493, A192494, A322029 (corresponding denominators).
%K A321693 nonn,frac,hard
%O A321693 3,1
%A A321693 _Hugo Pfoertner_, Nov 21 2018
%E A321693 a(21)-a(26) from _Hugo Pfoertner_, Dec 03 2018