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

1,1

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

Table of n, a(n) for n=1..49.

Gerhard Kirchner, Closest polygons around a circle

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

Cf. A357575, A357576.

Sequence in context: A229595 A083508 A048231 * A070169 A293410 A348270

Adjacent sequences: A357574 A357575 A357576 * A357578 A357579 A357580

Keyword

nonn

Author

Gerhard Kirchner, Oct 17 2022

Status

approved