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A373008
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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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1
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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
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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FORMULA
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{ 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 }.
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EXAMPLE
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Radius 2*r-2 rows 2*r-1 rows
19 1072 squares 1071 squares
52 8332 squares 8331 squares
65 13076 squares 13073 squares
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MATHEMATICA
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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[#] &]
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CROSSREFS
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Cf. A125228 (odd number of rows with maximum squares per row), A372847 (even number of rows with maximum squares per row).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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