A125228 Maximal number of squares of side 1 in a disk of radius n.

1, 7, 21, 39, 65, 93, 135, 179, 227, 285, 349, 415, 495, 573, 663, 759, 859, 963, 1071, 1199, 1325, 1457, 1591, 1735, 1891, 2049, 2207, 2383, 2557, 2735, 2929, 3123, 3327, 3529, 3739, 3955, 4191, 4427, 4665, 4901, 5159, 5413, 5681, 5951, 6231, 6515, 6799

Offset: 1

Author

Filippo ALUFFI PENTINI (falpen(AT)gmail.com), Jan 25 2007

Keywords

• easy
• nonn

Examples

			a(2)=7 since you cannot pack more than 7 unit-side squares in a disk of radius 2
		

Links

• David Dewan, Table of n, a(n) for n = 1..10000
• David Dewan, Drawings for n=1..12

Crossrefs

Similar to A001182 but less constrained.

A124484 is another version.

Programs

Mathematica

f[n_] := 2 Sum[ IntegerPart[2 Sqrt[n^2 - (n - k - 1/2)^2]], {k, 0, n - 2}] + IntegerPart[2 Sqrt[n^2 - 1/2^2]]; Array[f, 47] (* Robert G. Wilson v, Jan 27 2007 *)
a[n_]:=2 Sum[Floor[2 Sqrt[n^2-(k+1/2)^2]],{k,n-1}]+2n-1;
Array[a, 47]  (*  David Dewan, Jun 07 2024*)

Python

from math import isqrt
def A125228(n): return (m:=n<<1)-1+(sum(isqrt((k*(m-k+1)-n<<2)-1) for k in range(1,n))<<1) # Chai Wah Wu, Jul 18 2024

Formula

a(n) = 2*Sum_{k=1..n-1} floor(2*sqrt(n^2 - (k+1/2)^2)) + 2*n - 1.

Extensions

More terms from Robert G. Wilson v, Jan 27 2007