A357577 -id:A357577 - 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.

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

Gerhard Kirchner, Nov 18 2022

nonn

"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

			See link.

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

Cf. A357577.

A386538 a(n) is the maximum possible area of a polygon within a circle of radius n, where both the center and the vertices lie on points of a unit square grid.

Original entry on oeis.org

0, 2, 8, 24, 42, 74, 104, 138, 186, 240, 304, 362, 424, 512, 594, 690, 776, 880, 986, 1104, 1232, 1346, 1490, 1624, 1762, 1930, 2088, 2256, 2418, 2594, 2784, 2962, 3170, 3368, 3584, 3810, 4008, 4248, 4466, 4730, 4976, 5210, 5474, 5736, 6024, 6306, 6570, 6864, 7154

Offset: 0

Felix Huber, Aug 05 2025

nonn

a(n) > 99% of the circle area for n >= 50.
Conjecture: The maximum possible area of a polygon within the circle would be the same if only the vertices but not the center were fixed on grid points.
All terms are even.

			See linked illustration of the term a(4) = 42.

Felix Huber, Table of n, a(n) for n = 0..10000
Felix Huber, Illustration of a(4) = 42
Eric Weisstein's World of Mathematics, Pick's Theorem

Cf. A000217, A066643, A123690, A125228, A288247, A322106, A322107, A357577, A386539.

A386538:=proc(n)
    local x,y,p,s;
    p:=4*n;
    s:={};
    for x to n do
        y:=floor(sqrt(n^2-x^2));
        p:=p+4*y;
        s:=s union {y}
    od;
    return p-2*nops(s)
end proc;
seq(A386538(n),n=0..48);

a[n_] := Module[{p=4n},s = {}; Do[ y = Floor[Sqrt[n^2 - x^2]];p = p + 4*y;s = Union[s, {y}],{x,n} ];p - 2*Length[s]];Array[a,49,0] (* James C. McMahon, Aug 19 2025 *)

a(n) = A386539(A000217(n)) = A386539(n,n) for n >= 1.

a(n) <= A066643(n).

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.

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A386538 a(n) is the maximum possible area of a polygon within a circle of radius n, where both the center and the vertices lie on points of a unit square grid.

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