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%I #25 Aug 28 2016 18:23:40
%S 0,2,4,10,14,24,30,48,60,76,92,110,130,154,178,208,230,264,288,330,
%T 364,406,442,482,522,564,614,664,712,766,812,874,922,990,1050,1112,
%U 1176,1240,1312,1382,1452,1530,1598,1684,1750,1840,1920,2008,2092,2182,2266
%N Circle numbers (version 5): a(n) = number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (1/2,0).
%C Equivalently, number of points in a hexagonal lattice covered by a circular disk of diameter n if the center of the circle is chosen at the middle between two lattice points. - _Hugo Pfoertner_, Jan 07 2007
%C Same as above but "number of disks (r = 1)" instead of "number of points". a(2^n - 1) = A239073(n), n >= 1. See illustration in links. - _Kival Ngaokrajang_, Apr 06 2014
%H H. v. Eitzen, <a href="/A053417/b053417.txt">Table of n, a(n) for n = 0..1000</a>
%H Kival Ngaokrajang, <a href="/A053417/a053417_1.pdf">Illustration of initial terms</a>
%H <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>
%F a(n)/(n/2)^2 -> Pi*2/sqrt(3).
%t a[n_] := Sum[dj = Sqrt[Abs[4*n^2 + 6*i - 3*i^2 - 3]]/4; j1 = (1 - 2*i)/4 - dj // Floor; j2 = (1 - 2*i)/4 + dj // Ceiling; Sum[ Boole[i^2 - i - j/2 + i*j + j^2 + 1/4 <= n^2/4], {j, j1, j2}], {i, -n - 1, n + 3}]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jun 06 2013 *)
%Y Cf. A053411, A053414, A053415, A053416, A053417.
%Y Cf. A053479, A125851, A125852.
%Y Cf. A239073.
%K easy,nonn
%O 0,2
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 10 2000
%E Edited by _N. J. A. Sloane_, Jul 03 2008 at the suggestion of _R. J. Mathar_
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