A372847 -id:A372847 - cp's OEIS frontend

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.

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

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

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.

Views

Author

Keywords

Comments

Examples

Links

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

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.