A281795 Number of unit squares (partially) covered by a disk of radius n centered at the origin.

0, 4, 16, 36, 60, 88, 132, 172, 224, 284, 344, 416, 484, 568, 664, 756, 856, 956, 1076, 1200, 1324, 1452, 1600, 1740, 1884, 2040, 2212, 2392, 2560, 2732, 2928, 3120, 3332, 3536, 3748, 3980, 4192, 4428, 4660, 4920, 5172, 5412, 5688, 5956, 6248, 6528, 6804, 7104, 7400, 7716

Offset: 0

Author

Orson R. L. Peters, Jan 30 2017

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A001182, A242118.

Programs

Maple

N:= 100:  # for a(0)..a(N)
V:=Array(0..N):
for i from 0 to N do
  for j from 0 to i do
    r:= sqrt(i^2 + j^2);
    if r::integer then r:= r+1 else r:= ceil(r) fi;
    if r > N then break fi;
    if i=j then m:= 4 else m:= 8 fi;
    V[r..N]:= V[r..N] +~ m;
od od:
convert(V,list); # Robert Israel, Feb 21 2025

Mathematica

A281795[n_] := 4*Sum[Ceiling[Sqrt[n^2 - k^2]], {k, 0, n-1}];
Array[A281795, 100, 0] (* Paolo Xausa, Feb 21 2025 *)

Octave

a = @(n) 4*sum(ceil(sqrt(n.^2-(0:n-1).^2))); % Luis Mendo, Aug 09 2021

Python

a = lambda n: sum(4 for x in range(n) for y in range(n)
                    if x*x + y*y < n*n)

Formula

a(n) = 4*A001182(n) + A242118(n). - Andrey Zabolotskiy, Jan 30 2017

a(n) = Sum_{k=0..n-1} 4*ceiling(sqrt(n^2-k^2)). - Luis Mendo, Aug 09 2021