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

%I A357576 #16 Oct 23 2022 11:27:26
%S A357576 0,2,8,17,28,46,63,87,112,142,173,204,244,287,333,378,428,485,540,602,
%T A357576 661,737,802,869,947,1030,1118,1197,1278,1378,1469,1575,1670,1776,
%U A357576 1889,1990,2108,2219,2353,2472,2587,2723,2854,3002,3135,3275,3424,3563,3721
%N A357576 Half area of the convex hull of {(x,y)| x,y integers and x^2 + y^2 < n^2}.
%C A357576 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.
%H A357576 Gerhard Kirchner, <a href="/A357576/a357576.pdf">Examples for n <= 13</a>.
%F A357576 a(n) = A357575(n) - 2*floor(sqrt(2*n-1)) if n is a nonhypotenuse number (A004144).
%e A357576 For n=4: 5+6+6 = 17 square units -> a(4)=17.
%e A357576      _______
%e A357576    /|_|_|_|_|\  5
%e A357576   |_|_|_|_|_|_| 6
%e A357576   |_|_|_|_|_|_| 6
%o A357576 (Maxima)
%o A357576 block(nmax: 40, a: makelist(0,i,1,nmax), a[1]:0,
%o A357576 for n from 2 thru nmax do
%o A357576   (x0:0, y0:n, xa:0, ya:n, m1:0, m0:2, ar:0,
%o A357576     while xa<ya do (y:y0,
%o A357576      while m1<=m0 and xa<ya do
%o A357576       (y:y-1, x1: sqrt(n^2-y^2-1), m1: (y0-y)/(x1-x0),
%o A357576        if m1<=m0 then (x:floor(x1), m: (y0-y)/(x-x0),
%o A357576          if m<m0 then (m0:m, xa:x, ya:y))),
%o A357576       dar:xa*y0-ya*x0, if y0=n then dar:dar-xa,
%o A357576       if xa<=ya then (x0:xa, y0:ya, m0:2, ar:ar+2*dar) else ar:ar+dar),
%o A357576        a[n]: ar), a);
%o A357576 (Python)
%o A357576 from math import isqrt
%o A357576 from sympy import convex_hull
%o A357576 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
%Y A357576 Cf. A000328, A004144, A292276, A357575.
%K A357576 nonn
%O A357576 1,2
%A A357576 _Gerhard Kirchner_, Oct 05 2022