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A386538 a(n) is the maximum possible area of a polygon within a circle of radius n, where both the center and the vertices lie on points of a unit square grid.

%I A386538 #23 Aug 19 2025 09:44:07
%S A386538 0,2,8,24,42,74,104,138,186,240,304,362,424,512,594,690,776,880,986,
%T A386538 1104,1232,1346,1490,1624,1762,1930,2088,2256,2418,2594,2784,2962,
%U A386538 3170,3368,3584,3810,4008,4248,4466,4730,4976,5210,5474,5736,6024,6306,6570,6864,7154
%N A386538 a(n) is the maximum possible area of a polygon within a circle of radius n, where both the center and the vertices lie on points of a unit square grid.
%C A386538 a(n) > 99% of the circle area for n >= 50.
%C A386538 Conjecture: The maximum possible area of a polygon within the circle would be the same if only the vertices but not the center were fixed on grid points.
%C A386538 All terms are even.
%H A386538 Felix Huber, <a href="/A386538/b386538.txt">Table of n, a(n) for n = 0..10000</a>
%H A386538 Felix Huber, <a href="/A386538/a386538.pdf">Illustration of a(4) = 42</a>
%H A386538 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PicksTheorem.html">Pick's Theorem</a>
%F A386538 a(n) = A386539(A000217(n)) = A386539(n,n) for n >= 1.
%F A386538 a(n) <= A066643(n).
%e A386538 See linked illustration of the term a(4) = 42.
%p A386538 A386538:=proc(n)
%p A386538     local x,y,p,s;
%p A386538     p:=4*n;
%p A386538     s:={};
%p A386538     for x to n do
%p A386538         y:=floor(sqrt(n^2-x^2));
%p A386538         p:=p+4*y;
%p A386538         s:=s union {y}
%p A386538     od;
%p A386538     return p-2*nops(s)
%p A386538 end proc;
%p A386538 seq(A386538(n),n=0..48);
%t A386538 a[n_] := Module[{p=4n},s = {}; Do[ y = Floor[Sqrt[n^2 - x^2]];p = p + 4*y;s = Union[s, {y}],{x,n} ];p - 2*Length[s]];Array[a,49,0] (* _James C. McMahon_, Aug 19 2025 *)
%Y A386538 Cf. A000217, A066643, A123690, A125228, A288247, A322106, A322107, A357577, A386539.
%K A386538 nonn
%O A386538 0,2
%A A386538 _Felix Huber_, Aug 05 2025