A125228 -id:A125228 - cp's OEIS frontend

A372847 Number of unit squares enclosed by a circle of radius n with an even number of rows and the maximum number of squares in each row.

0, 6, 18, 36, 64, 92, 130, 172, 224, 284, 344, 410, 488, 570, 658, 750, 852, 956, 1072, 1194, 1312, 1450, 1584, 1728, 1882, 2044, 2204, 2372, 2548, 2730, 2916, 3112, 3312, 3520, 3738, 3950, 4184, 4408, 4656, 4900, 5146, 5402, 5670, 5942, 6222, 6492, 6784, 7080, 7382, 7700

Offset: 1

David Dewan, May 14 2024

nonn

Always has an even number of rows (2*n-2) and each row may have an odd or even number of squares.

Symmetrical about the horizontal and vertical axes.

			For n=4
row 1:   5 squares
row 2:   6 squares
row 3:   7 squares
row 4:   7 squares
row 5:   6 squares
row 6:   5 squares
Total = 36

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

Cf. A136485 (by diameter), A001182 (within quadrant), A136483 (quadrant by diameter), A119677 (even number of rows with even number of squares in each), A125228 (odd number of rows with maximal squares per row), A341198 (points rather than squares).

a[n_]:=2 Sum[Floor[2 Sqrt[n^2 - k^2]], {k,n-1}]; Array[a,50]

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

A373193 On a unit square grid, the number of squares enclosed by a circle of radius n with origin at the center of a square.

Original entry on oeis.org

1, 5, 21, 37, 61, 89, 129, 177, 221, 277, 341, 401, 489, 561, 657, 749, 845, 949, 1049, 1185, 1313, 1441, 1573, 1709, 1877, 2025, 2185, 2361, 2529, 2709, 2901, 3101, 3305, 3505, 3713, 3917, 4157, 4397, 4637, 4865, 5121, 5377, 5637, 5917, 6197, 6485, 6761

Offset: 1

David Dewan, May 27 2024

nonn

This corresponds to a circle of radius n with center at 1/2,1/2 on a unit square grid.
Always has an odd number of rows (2 n - 1) with an odd number of squares in each row.
Symmetrical about the horizontal and vertical axes.

			For n=4:
  row 1: 3 squares   - - X X X - -
  row 2: 5 squares   - X X X X X -
  row 3: 7 squares   X X X X X X X
  row 4: 7 squares   X X X X X X X
  row 5: 7 squares   X X X X X X X
  row 6: 5 squares   - X X X X X -
  row 7: 3 squares   - - X X X - -
Total = 37 = a(4).

David Dewan, Table of n, a(n) for n = 1..10000
David Dewan, Drawings for n=1..10

Cf. A119677 (on unit square grid with circle center at origin), A372847 (even number of rows with maximal squares per row), A125228 (odd number of rows with maximal squares per row), A000328 (number of squares whose centers are inside the circle).

Table[4*Sum[Floor[Sqrt[n^2-(k+1/2)^2]-1/2],{k,1,n-1}]+4*n-3,{n,50}]

a(n) = 4*Sum_{k=1..n-1} floor(sqrt(n^2 - (k+1/2)^2) - 1/2) + 4*n - 3.

a(n) == 1 (mod 4). - Robert FERREOL, Jan 31 2025

A386538 a(n) is the maximum possible area of a polygon within a circle of radius n, where both the center and the vertices lie on points of a unit square grid.

Original entry on oeis.org

0, 2, 8, 24, 42, 74, 104, 138, 186, 240, 304, 362, 424, 512, 594, 690, 776, 880, 986, 1104, 1232, 1346, 1490, 1624, 1762, 1930, 2088, 2256, 2418, 2594, 2784, 2962, 3170, 3368, 3584, 3810, 4008, 4248, 4466, 4730, 4976, 5210, 5474, 5736, 6024, 6306, 6570, 6864, 7154

Offset: 0

Felix Huber, Aug 05 2025

nonn

a(n) > 99% of the circle area for n >= 50.
Conjecture: The maximum possible area of a polygon within the circle would be the same if only the vertices but not the center were fixed on grid points.
All terms are even.

			See linked illustration of the term a(4) = 42.

Felix Huber, Table of n, a(n) for n = 0..10000
Felix Huber, Illustration of a(4) = 42
Eric Weisstein's World of Mathematics, Pick's Theorem

Cf. A000217, A066643, A123690, A125228, A288247, A322106, A322107, A357577, A386539.

A386538:=proc(n)
    local x,y,p,s;
    p:=4*n;
    s:={};
    for x to n do
        y:=floor(sqrt(n^2-x^2));
        p:=p+4*y;
        s:=s union {y}
    od;
    return p-2*nops(s)
end proc;
seq(A386538(n),n=0..48);

a[n_] := Module[{p=4n},s = {}; Do[ y = Floor[Sqrt[n^2 - x^2]];p = p + 4*y;s = Union[s, {y}],{x,n} ];p - 2*Length[s]];Array[a,49,0] (* James C. McMahon, Aug 19 2025 *)

a(n) = A386539(A000217(n)) = A386539(n,n) for n >= 1.

a(n) <= A066643(n).

A108126 Maximal number of squares of side 1 in an ellipse of semiaxes n,2n.

Original entry on oeis.org

3, 17, 43, 83, 137, 203, 279, 369, 471, 587, 715, 857, 1011, 1175, 1351, 1541, 1743, 1961, 2191, 2429, 2683, 2949, 3227, 3523, 3829, 4137, 4469, 4809, 5167, 5539, 5913, 6295, 6701, 7127, 7555, 7999, 8449, 8909, 9395, 9889, 10395, 10915

Offset: 1

Pasquale CUTOLO (p.cutolo(AT)inwind.it), Jun 14 2007

			a(1)=3 since you cannot pack more than 3 unit-side squares in an ellipse of semiaxes 1,2

Similar to A125228.

f[n_] := 2 Sum[IntegerPart[2 Sqrt[4 n^2 - (h - 1/2)^2]],
{h, 2, 2 n}] + IntegerPart[2 Sqrt[4 n^2 - 1/4]]; Array[f,42]

A373008 Radii r of circles that can enclose more unit squares when having fewer rows of squares: 2r - 2 rows instead of 2r - 1 rows.

Original entry on oeis.org

19, 52, 65, 184, 197, 222, 230, 303, 328, 341, 425, 489, 646, 985, 1018, 1328, 1383, 1400, 1637, 1743, 1806, 1870, 1938, 1997, 2060, 2065, 2179, 2192, 2433, 2603, 2610, 2611, 2675, 2692, 2747, 2895, 2925, 2975, 3008, 3107, 3254, 3446, 3462, 3619, 3635

Offset: 1

David Dewan, May 19 2024

nonn

Numbers r for which A372847(r) > A125228(r).

For circles with these radii, a smaller number of rows (2*r - 2) allows more efficient packing than a larger number of rows (2*r - 1).

			Radius     2*r-2 rows         2*r-1 rows
19          1072 squares       1071 squares
52          8332 squares       8331 squares
65         13076 squares      13073 squares

David Dewan, Table of n, a(n) for n = 1..93
David Dewan, Drawings for A373008.

Cf. A125228 (odd number of rows with maximum squares per row), A372847 (even number of rows with maximum squares per row).

lessRows[r_] := 2 Sum[Floor[2 Sqrt[r^2 - k^2]], {k, r - 1}]
moreRows[r_] := 2 Sum[Floor[2 Sqrt[r^2 - (k + 1/2)^2]], {k, r - 1}] + 2 r - 1
Select[Range@100,lessRows[#] > moreRows[#] &]

{ r : 2*Sum_{k=1..r-1} floor(2*sqrt(r^2 - k^2)) > 2*Sum_{k=1..r-1} floor(2*sqrt(r^2 - (k+1/2)^2)) + 2*r - 1 }.

cp's OEIS Frontend

A372847 Number of unit squares enclosed by a circle of radius n with an even number of rows and the maximum number of squares in each row.

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A373193 On a unit square grid, the number of squares enclosed by a circle of radius n with origin at the center of a square.

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A386538 a(n) is the maximum possible area of a polygon within a circle of radius n, where both the center and the vertices lie on points of a unit square grid.

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Maple

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A108126 Maximal number of squares of side 1 in an ellipse of semiaxes n,2n.

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A373008 Radii r of circles that can enclose more unit squares when having fewer rows of squares: 2*r - 2 rows instead of 2*r - 1 rows.

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Examples

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Crossrefs

Programs

Mathematica

Formula

A373008 Radii r of circles that can enclose more unit squares when having fewer rows of squares: 2r - 2 rows instead of 2r - 1 rows.