A358465 Least area (doubled) of a triangle enclosing a circle of radius n such that the center of the circle and the vertices of the triangle all have integer coordinates.

12, 45, 96, 168, 269, 380, 520, 670, 861, 1044, 1274, 1508, 1760, 2050, 2340, 2680, 3016, 3383, 3762, 4176, 4588, 5052, 5511, 6000, 6512, 7040, 7584, 8160, 8758, 9360, 10010, 10659, 11352, 12036, 12753, 13482, 14238, 15032, 15812, 16640, 17500, 18352, 19240, 20153, 21060

Offset: 1

Author

Gerhard Kirchner, Nov 18 2022

Keywords

• nonn

Comments

"Enclosing" means that each edge lies outside the circle or is tangent to it.

The area of a "grid triangle" with integer vertex coordinates is a multiple of 1/2. If (0,0) is the center of the circle, a grid triangle exists with a vertex (x0,y0), 0 <= x0 <= y0 (because of the grid symmetry) such that the area is minimized.

The basic idea of finding the minimum: Generate triangles with vertices (x0,y0), (x1,y1), (x2,y2) such that all edges are tangents and replace (x1,y1) and (x2,y2) with points with integer coordinates in the neighborhood.

Limit_{n->oo} a(n)/n^2 = 6*sqrt(3). - Jon E. Schoenfield, Nov 19 2022

Examples

			See link.
		

Links

• Gerhard Kirchner, Examples and algorithm
• Gerhard Kirchner, Visual Basic program
• Gerhard Kirchner, Diagrams

Crossrefs

Cf. A357577.