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%I #20 Nov 24 2021 11:34:22
%S 0,0,4,4,12,16,32,32,52,60,80,88,112,124,156,172,208,216,256,276,316,
%T 332,384,408,448,484,540,560,616,648,716,740,812,848,912,952,1020,
%U 1060,1124,1184,1264,1304,1396,1436,1528,1576,1664,1716,1804,1876,1976
%N Circle numbers (version 3): a(n) = number of points (i,j), i,j integers, contained in a circle of diameter n, centered at (1/2, 1/2).
%H G. C. Greubel, <a href="/A053415/b053415.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n)/(n/2)^2 -> Pi.
%F a(n) = [x^(n^2)] theta_2(x^4)^2 / (1 - x). - _Ilya Gutkovskiy_, Nov 23 2021
%t cx = 1/2; cy = 1/2; a[n_] := Sum[ dj = 1/2*Sqrt[ Abs[n^2 - 4*cx^2 + 8*cx*i - 4*i^2]]; j1 = cy - dj // Floor; j2 = cy + dj // Ceiling; Sum[Boole[ (i - cx)^2 + (j - cy)^2 <= n^2/4], {j, j1, j2}], {i, cx - n/2 // Floor, cx + n/2 // Ceiling}]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jun 06 2013 *)
%Y Cf. A053411, A053414, A053415, A053416, A053417.
%K easy,nonn
%O 0,3
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 10 2000