A358681 Largest area (doubled) of a triangle enclosed by a circle of radius n such that the center of the circle and the vertices of the triangle all have integer coordinates.

2, 8, 21, 36, 64, 90, 120, 157, 208, 256, 306, 360, 432, 504, 576, 650, 750, 832, 928, 1025, 1122, 1254, 1360, 1480, 1612, 1748, 1886, 2016, 2170, 2328, 2484, 2646, 2802, 3000, 3180, 3348, 3540, 3718, 3944, 4148, 4340, 4552, 4788, 5016, 5244, 5473, 5718, 5964

Offset

1,1

Comments

"Enclosing" means that all vertices are located on or inside the circle. (0,0) is the center of the circle. The area of a "grid triangle" with integer vertex coordinates is a multiple of 1/2. The vertices of any largest triangle are "close points", i.e., points (x,y) on or inside the circle such that at least two of the four points (x+-1,y+-1) are located outside (see link).

There are terms b(n) and c(n) with b(n) <= a(n) <= c(n), see formula section. For any n, a triangle with the doubled area b(n) exists. By checking b(n)=c(n), one obtains:

a( 1) = 2

a( 5) = 64

a( 17) = 750

a( 65) = 10976

a( 241) = 150898

a( 901) = 2109120

a( 3361) = 29348702

a( 12545) = 408877504

a( 46817) = 5694545250

a(174725) = 79316215616

a(652081) = 1104727025998

The sequence of indices is A120893.

Links

Table of n, a(n) for n=1..48.

Gerhard Kirchner, Examples and algorithm

Formula

Limits: b(n) <= a(n) <= c(n) with

c(n) = floor(n ^ 2 * sqrt(3) * 3 / 2).

With f(n,t) = (3*n + t) * floor(sqrt((3*n + t) * (n - t)) / 2):

b(n) = f(n,0) for even n

b(n) = max{f(n,-1),f(n,1)} for odd n.

a(n)/n^2 tends to sqrt(3) * 3 / 2.

Example

see link

Prog

(Maxima)
block(nmax: 50, a: makelist(0, i, 1, nmax),
for n from 1 thru nmax do
  (p: floor(n/sqrt(2)), if 2*p*(p+1) > n^2-1 then d:0 else d:1,
   k0:n+1-d-p, z:2*k0-2+d,
   v: makelist([0, 0], i, 1, 4*z), ma:0,
        /*Find close points with y>=0, x<=y:*/
   for k from 1 thru k0 do (j:k-1, v[k][2]:n-j, v[k][1]: floor(sqrt(j*(2*n-j)))),
          /*mirror x=y:*/
   for k from k0+d thru z+1 do(j:z+2-k, v[k][1]: v[j][2], v[k][2]: v[j][1]),
          /*rotation 90°:*/
   for k from z+2 thru 4*z do(j:k-z, v[k][1]: v[j][2], v[k][2]: -v[j][1]),
          /*All 4*z close points found, generate triangles:*/
   for k from 1 thru k0 do
    for j from k+2*z+1 thru k+3*z do
     for m from k+z thru k+2*z-1 do
      (ar:(v[j][1]-v[k][1])*(v[m][2]-v[k][2])-(v[j][2]-v[k][2])*(v[m][1]-v[k][1]),
       if ar>ma then ma:ar),  a[n]:ma), a);

Crossrefs

Cf. A358465, A120893.

Sequence in context: A001471 A162585 A000159 * A090612 A355760 A212981

Adjacent sequences: A358678 A358679 A358680 * A358682 A358683 A358684

Keyword

nonn

Author

Gerhard Kirchner, Nov 26 2022

Status

approved