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

0,3

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.

Links

Table of n, a(n) for n=0..49.

Gerhard Kirchner, Examples for n <= 13

Example

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

Prog

(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<ya do (y:y0,
   while m1<=m0 and xa<ya do
    (y:y-1, x1: sqrt(n^2-y^2), m1: (y0-y)/(x1-x0),
     if m1<=m0 then (x:floor(x1), m: (y0-y)/(x-x0),
      if m<m0 then (m0:m, xa:x, ya:y))),
       dar:xa*y0-ya*x0,
       if xa<=ya then (x0:xa, y0:ya, m0:2, ar:ar+2*dar) else ar:ar+dar),
  a[n]: ar), a);
(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

Crossrefs

Cf. A357576, A292276.

Sequence in context: A008167 A008150 A301234 * A212965 A301226 A008200

Adjacent sequences: A357572 A357573 A357574 * A357576 A357577 A357578

Keyword

nonn

Author

Gerhard Kirchner, Oct 04 2022

Status

approved