A124484 -id:A124484 - cp's OEIS frontend

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

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

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

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

Similar to A001182 but less constrained.

A124484 is another version.

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*)

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

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

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

A374505 Maximum number of unit squares aligned with unit-spaced horizontal lines that can be enclosed by a circle of diameter n.

Original entry on oeis.org

0, 0, 1, 4, 8, 14, 21, 29, 40, 52, 65, 81, 97, 116, 135, 156, 180, 203, 229, 258, 286, 317, 350, 383, 419, 455, 495, 536, 575, 620, 664, 711, 761, 808, 860, 916, 966, 1024, 1079, 1140, 1200, 1261, 1326, 1391, 1458, 1528, 1595, 1666, 1741, 1814, 1892, 1972

Offset: 0

David Dewan, Jul 09 2024

nonn

It is conjectured that this construction gives the maximal number of axis-parallel unit squares that can be packed into a circle of diameter n.

From the Erich Friedman website the best known maximum number of unit squares enclosed by a circle of diameter n are for n >= 2: 1, 4, 8, 14, 21, 30, ... (this sequence has not been included in OEIS because the terms have not been proven optimal). The unit squares in this case are not required to be axis-parallel. However, the example of 30 axis-parallel squares enclosed in a circle of radius < 3.5 shows that if holes are allowed, better packings are possible than with the restricted case. - Andrew Howroyd, Jul 14 2024

			For a circle with diameter = 4:
With center of circle at y = 0 (on line between rows) it encloses 6 squares.
With center of circle at y = 2 - sqrt(3) ~= 0.268 it encloses 8 squares (maximal).
With center of circle at y = 1/2 (in middle of row) it encloses 7 squares.
So a(4) = 8.

David Dewan, Table of n, a(n) for n = 0..2000
David Dewan, Computing Maximal Unit Squares in a Circle.
Erich Friedman, Squares in Circles.

Cf. A124484, A256588 (unexpectedly similar).

a[n_] := (
  distances = N[Map[Sqrt[n^2 - #^2]/2 &, Range[n - 1]]];
  topDeltas1 = Flatten[Map[# - distances &, Range[n/2]]];
  topDeltas2 = Select[topDeltas1, 0 < # <= .5 &];
  topDeltas3 = Map[{#, 1} &, topDeltas2];
  btmDeltas1 = Flatten[Map[distances - # &, Range[n/2]]];
  btmDeltas2 = Select[btmDeltas1, 0 <= # < .5 &];
  btmDeltas3 = Map[{#, -1} &, btmDeltas2];
  allDeltas4 = Join[topDeltas3, btmDeltas3, {{0, 0}}];
  allDeltas5 = SortBy[allDeltas4, {First, -Last[#] &}] ;
  cumulativeChanges = Accumulate[allDeltas5[[All, 2]]];
  startSqrs = 2 Sum[Floor[2 Sqrt[(n/2)^2 - k^2]], {k, n/2}];
  Return[startSqrs + Max[cumulativeChanges]]  )
Map[a[#] &, Range[0, 51]]      (* this sequence *)
Map[a[#] &, Range[0, 102, 2]]  (* A124484, by radius *)

a(2*n) <= A124484(n).

A374528 Conjectured number of unit squares setting a new density record when packed into the smallest possible enclosing circle.

Original entry on oeis.org

1, 6, 7, 10, 11, 12, 19, 21, 24, 26, 27

Offset: 1

Hugo Pfoertner, Jul 16 2024

All data after 7 have to be considered as conjectural. The sequence is based on the configurations shown on Erich Friedman's "Squares in Circles" webpage, see link.

Erich Friedman, Squares in Circles.
Montanher, T., Neumaier, A., Csaba Markót, M. et al., Rigorous packing of unit squares into a circle. J Glob Optim 73, 547-565 (2019).
Hugo Pfoertner, Data of packings, from E. Friedman's webpage as of Jul 17, 2024.

Cf. A124484.

cp's OEIS Frontend

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

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A374505 Maximum number of unit squares aligned with unit-spaced horizontal lines that can be enclosed by a circle of diameter n.

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Examples

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Programs

Mathematica

Formula

A374528 Conjectured number of unit squares setting a new density record when packed into the smallest possible enclosing circle.

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Author

Keywords

Comments

Links

Crossrefs