cp's OEIS Frontend

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.

%I A358465 #28 Mar 02 2024 12:34:55
%S A358465 12,45,96,168,269,380,520,670,861,1044,1274,1508,1760,2050,2340,2680,
%T A358465 3016,3383,3762,4176,4588,5052,5511,6000,6512,7040,7584,8160,8758,
%U A358465 9360,10010,10659,11352,12036,12753,13482,14238,15032,15812,16640,17500,18352,19240,20153,21060
%N 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.
%C A358465 "Enclosing" means that each edge lies outside the circle or is tangent to it.
%C A358465 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.
%C A358465 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.
%C A358465 Limit_{n->oo} a(n)/n^2 = 6*sqrt(3). - _Jon E. Schoenfield_, Nov 19 2022
%H A358465 Gerhard Kirchner, <a href="/A358465/a358465.pdf">Examples and algorithm</a>
%H A358465 Gerhard Kirchner, <a href="/A358465/a358465_1.txt">Visual Basic program</a>
%H A358465 Gerhard Kirchner, <a href="/A358465/a358465_1.pdf">Diagrams</a>
%e A358465 See link.
%o A358465 (Visual Basic) ' See links.
%Y A358465 Cf. A357577.
%K A358465 nonn
%O A358465 1,1
%A A358465 _Gerhard Kirchner_, Nov 18 2022