A357576 Half area of the convex hull of {(x,y)| x,y integers and x^2 + y^2 < n^2}.

0, 2, 8, 17, 28, 46, 63, 87, 112, 142, 173, 204, 244, 287, 333, 378, 428, 485, 540, 602, 661, 737, 802, 869, 947, 1030, 1118, 1197, 1278, 1378, 1469, 1575, 1670, 1776, 1889, 1990, 2108, 2219, 2353, 2472, 2587, 2723, 2854, 3002, 3135, 3275, 3424, 3563, 3721

Offset

1,2

Comments

a(n) is odd if there is an edge connecting two vertices (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. 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 vertex coordinates are integers is a multiple of 1/2.

Links

Table of n, a(n) for n=1..49.

Gerhard Kirchner, Examples for n <= 13.

Formula

a(n) = A357575(n) - 2*floor(sqrt(2*n-1)) if n is a nonhypotenuse number (A004144).

Example

For n=4: 5+6+6 = 17 square units -> a(4)=17.
     _______
   /|_|_|_|_|\  5
  |_|_|_|_|_|_| 6
  |_|_|_|_|_|_| 6

Prog

(Maxima)
block(nmax: 40, a: makelist(0, i, 1, nmax), a[1]:0,
for n from 2 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^2-1), 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 y0=n then dar:dar-xa,
      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 A357576(n): return 0 if n == 1 else int(2*convex_hull(*[(0, 0), (n-1, 0)]+[(x, isqrt((n-x)*(n+x)-1)) for x in range(n)]).area) # Chai Wah Wu, Oct 23 2022

Crossrefs

Cf. A000328, A004144, A292276, A357575.

Sequence in context: A224865 A192159 A066564 * A034972 A294537 A294548

Adjacent sequences: A357573 A357574 A357575 * A357577 A357578 A357579

Keyword

nonn

Author

Gerhard Kirchner, Oct 05 2022

Status

approved