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.

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

Author

David Dewan, May 19 2024

Keywords

• nonn

Comments

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).

Examples

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

Links

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

Crossrefs

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

Programs

Mathematica

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[#] &]

Formula

{ 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 }.