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

Author

Gerhard Kirchner, Oct 04 2022

Keywords

• nonn

Comments

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.

Examples

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

Links

• Gerhard Kirchner, Examples for n <= 13

Crossrefs

Cf. A357576, A292276.

Programs

Maxima

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
				

Python

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