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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.
1
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
1,1
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).
LINKS
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 }.
EXAMPLE
Radius 2*r-2 rows 2*r-1 rows
19 1072 squares 1071 squares
52 8332 squares 8331 squares
65 13076 squares 13073 squares
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[#] &]
CROSSREFS
Cf. A125228 (odd number of rows with maximum squares per row), A372847 (even number of rows with maximum squares per row).
Sequence in context: A267572 A349554 A190267 * A066775 A118591 A031341
KEYWORD
nonn
AUTHOR
David Dewan, May 19 2024
STATUS
approved