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, 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

Offset: 1

Author

Gerhard Kirchner, Oct 17 2022

Keywords

• nonn

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).

Examples

			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".
		

Links

• Gerhard Kirchner, Closest polygons around a circle

Crossrefs

Cf. A357575, A357576.