A357575 - OEIS

%I #24 Oct 23 2022 20:46:34

%S 0,1,4,12,21,37,52,69,93,120,152,181,212,258,297,345,388,444,495,552,

%T 616,673,749,814,881,965,1046,1132,1211,1301,1396,1483,1589,1686,1800,

%U 1907,2006,2128,2235,2371,2490,2607,2741,2872,3020,3155,3293,3442,3581,3739

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

%C 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 Gerhard Kirchner, <a href="/A357575/a357575.pdf">Examples for n <= 13</a>

%e n=2: 1+3 square units -> a(2) = 4.

%e           ^

%e         / | \ 1

%e       /___|___\

%e     / |   |   | \ 3

%e   /___|___|___|___\

%o (Maxima)

%o block(nmax: 60, a: makelist(0,i,1,nmax),

%o for n from 1 thru nmax do

%o (x0:0, y0:n, xa:0, ya:n, m1:0, m0:2, ar:0,

%o   while xa<ya do (y:y0,

%o    while m1<=m0 and xa<ya do

%o     (y:y-1, x1: sqrt(n^2-y^2), m1: (y0-y)/(x1-x0),

%o      if m1<=m0 then (x:floor(x1), m: (y0-y)/(x-x0),

%o       if m<m0 then (m0:m, xa:x, ya:y))),

%o        dar:xa*y0-ya*x0,

%o        if xa<=ya then (x0:xa, y0:ya, m0:2, ar:ar+2*dar) else ar:ar+dar),

%o   a[n]: ar), a);

%o (Python)

%o from math import isqrt

%o from sympy import convex_hull

%o 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 Cf. A357576, A292276.

%K nonn

%O 0,3

%A _Gerhard Kirchner_, Oct 04 2022