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).

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

Links

H. v. Eitzen, Table of n, a(n) for n = 0..1000

Kival Ngaokrajang, Illustration of initial terms

Index entries for sequences related to A2 = hexagonal = triangular lattice

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. A003215, A125849, A125850, A125851, A125852.

Cf. A053411, A053414, A053415, A053417, A053458 (open disk), A308685 (bisection).

Sequence in context: A070919 A070847 A195863 * A213031 A299453 A300091

Adjacent sequences: A053413 A053414 A053415 * A053417 A053418 A053419

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