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A053416 Circle numbers (version 4): 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 (0,0).

%I A053416 #30 Nov 22 2022 10:46:47
%S A053416 1,1,7,7,19,19,37,43,61,73,91,109,127,151,187,199,241,253,301,313,367,
%T A053416 397,439,475,517,571,613,661,721,757,823,859,931,979,1045,1111,1165,
%U A053416 1237,1303,1381,1459,1519,1615,1663,1765,1813,1921,1993,2083,2173,2263
%N A053416 Circle numbers (version 4): 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 (0,0).
%C A053416 In other words, number of points in a hexagonal lattice covered by a circle of diameter n if the center of the circle is chosen at a grid point. - _Hugo Pfoertner_, Jan 07 2007
%C A053416 Same as above but "number of disks (r = 1)" instead of "number of points". See illustration in links. - _Kival Ngaokrajang_, Apr 06 2014
%H A053416 H. v. Eitzen, <a href="/A053416/b053416.txt">Table of n, a(n) for n = 0..1000</a>
%H A053416 Kival Ngaokrajang, <a href="/A053416/a053416_1.pdf">Illustration of initial terms</a>
%H A053416 <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>
%F A053416 a(n)/(n/2)^2->Pi*2/sqrt(3).
%F A053416 a(n) >= A053458(n). - _R. J. Mathar_, Nov 22 2022
%F A053416 a(2*n) = A308685(n). - _R. J. Mathar_, Nov 22 2022
%p A053416 A053416 := proc(d)
%p A053416     local a,j,imin,imax ;
%p A053416     a := 0 ;
%p A053416     for j from -floor(d/sqrt(3)) do
%p A053416         if j^2*3 > d^2 and j> 0 then
%p A053416             break ;
%p A053416         end if;
%p A053416         imin := ceil((-j-sqrt(d^2-3*j^2))/2) ;
%p A053416         imax := floor((-j+sqrt(d^2-3*j^2))/2) ;
%p A053416         a := a+imax-imin+1 ;
%p A053416     end do:
%p A053416     a ;
%p A053416 end proc:
%p A053416 seq(A053416(d),d=0..30) ; # _R. J. Mathar_, Nov 22 2022
%t A053416 a[n_] := Sum[Boole[4*(i^2 + i*j + j^2) <= n^2], {i, -n, n}, {j, -n, n}];
%t A053416 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jun 06 2013, updated Apr 08 2022 to correct a discrepancy wrt b-file noticed by Georg Fischer *)
%Y A053416 Cf. A003215, A125849, A125850, A125851, A125852.
%Y A053416 Cf. A053411, A053414, A053415, A053417, A053458 (open disk), A308685 (bisection).
%K A053416 easy,nonn
%O A053416 0,3
%A A053416 Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 10 2000
%E A053416 Edited by _N. J. A. Sloane_, Jul 03 2008 at the suggestion of _R. J. Mathar_