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Least half area of a convex polygon enclosing a circle with radius n and center (0,0) such that all vertex coordinates are integers.
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%I #14 Mar 02 2024 12:27:58

%S 2,7,16,26,42,59,80,104,132,163,194,229,274,312,360,406,465,516,573,

%T 637,698,772,838,910,993,1073,1158,1238,1333,1425,1520,1621,1719,1835,

%U 1936,2043,2165,2280,2405,2525,2650,2782,2919,3059,3195,3340,3486,3632,3786

%N Least half area of a convex polygon enclosing a circle with radius n and center (0,0) such that all vertex coordinates are integers.

%C "Enclosing" means that any edge runs outside the circle or is a tangent.

%C Such a polygon does not need to be symmetrical, but the partial areas in the four quadrants are equal. Therefore it is sufficient to find the least area of a quarter polygon (multiplied by 2). The half area is an integer because the area of any convex polygon whose vertex coordinates are integers is a multiple of 1/2. The least number of polygons minimizing the area is 16 if x=y is not an axis of symmetry (2 solutions for each quadrant).

%H Gerhard Kirchner, <a href="/A357577/a357577_1.pdf">Closest polygons around a circle</a>

%e For n=1: 2 X 2 square: a(1) = 4/2 = 2.

%e For n=2: Octagon with vertices (1,2) and (2,1) in the first quadrant: a(2) = 14/2 = 7.

%e For further examples, see "Closest polygons around a circle".

%o (Visual Basic) ' See "Closest polygons around a circle"

%Y Cf. A357575, A357576.

%K nonn

%O 1,1

%A _Gerhard Kirchner_, Oct 17 2022