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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).
15
1, 1, 7, 7, 19, 19, 37, 43, 61, 73, 91, 109, 127, 151, 187, 199, 241, 253, 301, 313, 367, 397, 439, 475, 517, 571, 613, 661, 721, 757, 823, 859, 931, 979, 1045, 1111, 1165, 1237, 1303, 1381, 1459, 1519, 1615, 1663, 1765, 1813, 1921, 1993, 2083, 2173, 2263
OFFSET
0,3
COMMENTS
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
Same as above but "number of disks (r = 1)" instead of "number of points". See illustration in links. - Kival Ngaokrajang, Apr 06 2014
FORMULA
a(n)/(n/2)^2->Pi*2/sqrt(3).
a(n) >= A053458(n). - R. J. Mathar, Nov 22 2022
a(2*n) = A308685(n). - R. J. Mathar, Nov 22 2022
MAPLE
A053416 := proc(d)
local a, j, imin, imax ;
a := 0 ;
for j from -floor(d/sqrt(3)) do
if j^2*3 > d^2 and j> 0 then
break ;
end if;
imin := ceil((-j-sqrt(d^2-3*j^2))/2) ;
imax := floor((-j+sqrt(d^2-3*j^2))/2) ;
a := a+imax-imin+1 ;
end do:
a ;
end proc:
seq(A053416(d), d=0..30) ; # R. J. Mathar, Nov 22 2022
MATHEMATICA
a[n_] := Sum[Boole[4*(i^2 + i*j + j^2) <= n^2], {i, -n, n}, {j, -n, n}];
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 *)
CROSSREFS
Cf. A053411, A053414, A053415, A053417, A053458 (open disk), A308685 (bisection).
Sequence in context: A070919 A070847 A195863 * A213031 A299453 A300091
KEYWORD
easy,nonn
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 10 2000
EXTENSIONS
Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar
STATUS
approved