%I #32 Feb 22 2025 09:57:02
%S 0,4,16,36,60,88,132,172,224,284,344,416,484,568,664,756,856,956,1076,
%T 1200,1324,1452,1600,1740,1884,2040,2212,2392,2560,2732,2928,3120,
%U 3332,3536,3748,3980,4192,4428,4660,4920,5172,5412,5688,5956,6248,6528,6804,7104,7400,7716
%N Number of unit squares (partially) covered by a disk of radius n centered at the origin.
%C Touching a unit square does not count as covering. E.g., the disk with radius 5 does not cover the unit square with (3, 4) as bottom-left corner.
%H Robert Israel, <a href="/A281795/b281795.txt">Table of n, a(n) for n = 0..2000</a>
%F a(n) = 4*A001182(n) + A242118(n). - _Andrey Zabolotskiy_, Jan 30 2017
%F a(n) = Sum_{k=0..n-1} 4*ceiling(sqrt(n^2-k^2)). - _Luis Mendo_, Aug 09 2021
%e a(4) = 4 * 15 = 60 because in the positive quadrant 15 unit squares are covered and the problem is symmetrical. In the bounding box of the circle only the unit squares in the corners are not (partially) covered, so a(4) = 8*8 - 4 = 60.
%p N:= 100: # for a(0)..a(N)
%p V:=Array(0..N):
%p for i from 0 to N do
%p for j from 0 to i do
%p r:= sqrt(i^2 + j^2);
%p if r::integer then r:= r+1 else r:= ceil(r) fi;
%p if r > N then break fi;
%p if i=j then m:= 4 else m:= 8 fi;
%p V[r..N]:= V[r..N] +~ m;
%p od od:
%p convert(V,list); # _Robert Israel_, Feb 21 2025
%t A281795[n_] := 4*Sum[Ceiling[Sqrt[n^2 - k^2]], {k, 0, n-1}];
%t Array[A281795, 100, 0] (* _Paolo Xausa_, Feb 21 2025 *)
%o (Python)
%o a = lambda n: sum(4 for x in range(n) for y in range(n)
%o if x*x + y*y < n*n)
%o (Octave)
%o a = @(n) 4*sum(ceil(sqrt(n.^2-(0:n-1).^2))); % _Luis Mendo_, Aug 09 2021
%Y Cf. A001182, A242118.
%K nonn,easy,changed
%O 0,2
%A _Orson R. L. Peters_, Jan 30 2017