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

0,2

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.

Links

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

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

Example

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.

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 *)

Prog

(Python)
a = lambda n: sum(4 for x in range(n) for y in range(n)
                    if x*x + y*y < n*n)
(Octave)
a = @(n) 4*sum(ceil(sqrt(n.^2-(0:n-1).^2))); % Luis Mendo, Aug 09 2021

Crossrefs

Cf. A001182, A242118.

Sequence in context: A014727 A174597 A044065 * A063540 A349223 A055808

Adjacent sequences: A281792 A281793 A281794 * A281796 A281797 A281798

Keyword

nonn,easy

Author

Orson R. L. Peters, Jan 30 2017

Status

approved