

A357577


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


2



2, 7, 16, 26, 42, 59, 80, 104, 132, 163, 194, 229, 274, 312, 360, 406, 465, 516, 573, 637, 698, 772, 838, 910, 993, 1073, 1158, 1238, 1333, 1425, 1520, 1621, 1719, 1835, 1936, 2043, 2165, 2280, 2405, 2525, 2650, 2782, 2919, 3059, 3195, 3340, 3486, 3632, 3786
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OFFSET

1,1


COMMENTS

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


LINKS



EXAMPLE

For n=1: 2 X 2 square: a(1) = 4/2 = 2.
For n=2: Octagon with vertices (1,2) and (2,1) in the first quadrant: a(2) = 14/2 = 7.
For further examples, see "Closest polygons around a circle".


PROG

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


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



