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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.

%I A373008 #19 May 22 2024 06:24:58
%S A373008 19,52,65,184,197,222,230,303,328,341,425,489,646,985,1018,1328,1383,
%T A373008 1400,1637,1743,1806,1870,1938,1997,2060,2065,2179,2192,2433,2603,
%U A373008 2610,2611,2675,2692,2747,2895,2925,2975,3008,3107,3254,3446,3462,3619,3635
%N 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.
%C A373008 Numbers r for which A372847(r) > A125228(r).
%C A373008 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).
%H A373008 David Dewan, <a href="/A373008/b373008.txt">Table of n, a(n) for n = 1..93</a>
%H A373008 David Dewan, <a href="/A373008/a373008.pdf">Drawings for A373008</a>.
%F A373008 { 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 }.
%e A373008 Radius     2*r-2 rows         2*r-1 rows
%e A373008 19          1072 squares       1071 squares
%e A373008 52          8332 squares       8331 squares
%e A373008 65         13076 squares      13073 squares
%t A373008 lessRows[r_] := 2 Sum[Floor[2 Sqrt[r^2 - k^2]], {k, r - 1}]
%t A373008 moreRows[r_] := 2 Sum[Floor[2 Sqrt[r^2 - (k + 1/2)^2]], {k, r - 1}] + 2 r - 1
%t A373008 Select[Range@100,lessRows[#] > moreRows[#] &]
%Y A373008 Cf. A125228 (odd number of rows with maximum squares per row), A372847 (even number of rows with maximum squares per row).
%K A373008 nonn
%O A373008 1,1
%A A373008 _David Dewan_, May 19 2024