A358465 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..45.

Gerhard Kirchner, Examples and algorithm

Gerhard Kirchner, Visual Basic program

Gerhard Kirchner, Diagrams

EXAMPLE

See link.

PROG

(Visual Basic) ' See links.

CROSSREFS

Cf. A357577.