A357576 -id:A357576 - cp's OEIS frontend

A357575 Half area of the convex hull of {(x,y) | x,y integers and x^2 + y^2 <= n^2}.

0, 1, 4, 12, 21, 37, 52, 69, 93, 120, 152, 181, 212, 258, 297, 345, 388, 444, 495, 552, 616, 673, 749, 814, 881, 965, 1046, 1132, 1211, 1301, 1396, 1483, 1589, 1686, 1800, 1907, 2006, 2128, 2235, 2371, 2490, 2607, 2741, 2872, 3020, 3155, 3293, 3442, 3581, 3739

Offset: 0

Gerhard Kirchner, Oct 04 2022

nonn

a(n) is odd if there is an edge connecting two corners (x,y) and (y,x), x > y > 0, such that x-y is odd. Otherwise, a(n) is even. a(n)/n^2 is not monotonous but tends to Pi/2. Apparently, a recurrence or another formula for a(n) does not exist. The convex hull has four symmetry axes: x=0, y=0, y=x, y=-x. Therefore it is sufficient to find the least area of a quarter polygon (multiplied by 2). The half area is an integer because the area of any convex polygon whose corner coordinates are integers is a multiple of 1/2.

			n=2: 1+3 square units -> a(2) = 4.
          ^
        / | \ 1
      /___|___\
    / |   |   | \ 3
  /___|___|___|___\

Gerhard Kirchner, Examples for n <= 13

Cf. A357576, A292276.

block(nmax: 60, a: makelist(0,i,1,nmax),
for n from 1 thru nmax do
(x0:0, y0:n, xa:0, ya:n, m1:0, m0:2, ar:0,
  while xa

from math import isqrt
from sympy import convex_hull
def A357575(n): return int(2*convex_hull(*[(n,0),(0, 0)]+[(x, isqrt((n-x)*(n+x))) for x in range(n)]).area) if n else 0 # Chai Wah Wu, Oct 23 2022

A357577 Least half area of a convex polygon enclosing a circle with radius n and center (0,0) such that all vertex coordinates are integers.

Original entry on oeis.org

2, 7, 16, 26, 42, 59, 80, 104, 132, 163, 194, 229, 274, 312, 360, 406, 465, 516, 573, 637, 698, 772, 838, 910, 993, 1073, 1158, 1238, 1333, 1425, 1520, 1621, 1719, 1835, 1936, 2043, 2165, 2280, 2405, 2525, 2650, 2782, 2919, 3059, 3195, 3340, 3486, 3632, 3786

Offset: 1

Gerhard Kirchner, Oct 17 2022

nonn

"Enclosing" means that any edge runs outside the circle or is a tangent.

Such a polygon does not need to be symmetrical, but the partial areas in the four quadrants are equal. Therefore it is sufficient to find the least area of a quarter polygon (multiplied by 2). The half area is an integer because the area of any convex polygon whose vertex coordinates are integers is a multiple of 1/2. The least number of polygons minimizing the area is 16 if x=y is not an axis of symmetry (2 solutions for each quadrant).

			For n=1: 2 X 2 square: a(1) = 4/2 = 2.
For n=2: Octagon with vertices (1,2) and (2,1) in the first quadrant: a(2) = 14/2 = 7.
For further examples, see "Closest polygons around a circle".

Gerhard Kirchner, Closest polygons around a circle

Cf. A357575, A357576.

cp's OEIS Frontend

A357575 Half area of the convex hull of {(x,y) | x,y integers and x^2 + y^2 <= n^2}.

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