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

%I A357575 #24 Oct 23 2022 20:46:34
%S A357575 0,1,4,12,21,37,52,69,93,120,152,181,212,258,297,345,388,444,495,552,
%T A357575 616,673,749,814,881,965,1046,1132,1211,1301,1396,1483,1589,1686,1800,
%U A357575 1907,2006,2128,2235,2371,2490,2607,2741,2872,3020,3155,3293,3442,3581,3739
%N A357575 Half area of the convex hull of {(x,y) | x,y integers and x^2 + y^2 <= n^2}.
%C A357575 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.
%H A357575 Gerhard Kirchner, <a href="/A357575/a357575.pdf">Examples for n <= 13</a>
%e A357575 n=2: 1+3 square units -> a(2) = 4.
%e A357575           ^
%e A357575         / | \ 1
%e A357575       /___|___\
%e A357575     / |   |   | \ 3
%e A357575   /___|___|___|___\
%o A357575 (Maxima)
%o A357575 block(nmax: 60, a: makelist(0,i,1,nmax),
%o A357575 for n from 1 thru nmax do
%o A357575 (x0:0, y0:n, xa:0, ya:n, m1:0, m0:2, ar:0,
%o A357575   while xa<ya do (y:y0,
%o A357575    while m1<=m0 and xa<ya do
%o A357575     (y:y-1, x1: sqrt(n^2-y^2), m1: (y0-y)/(x1-x0),
%o A357575      if m1<=m0 then (x:floor(x1), m: (y0-y)/(x-x0),
%o A357575       if m<m0 then (m0:m, xa:x, ya:y))),
%o A357575        dar:xa*y0-ya*x0,
%o A357575        if xa<=ya then (x0:xa, y0:ya, m0:2, ar:ar+2*dar) else ar:ar+dar),
%o A357575   a[n]: ar), a);
%o A357575 (Python)
%o A357575 from math import isqrt
%o A357575 from sympy import convex_hull
%o A357575 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
%Y A357575 Cf. A357576, A292276.
%K A357575 nonn
%O A357575 0,3
%A A357575 _Gerhard Kirchner_, Oct 04 2022