A053458 Open disk numbers (version 3): a(n) is the number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in an open disk of diameter n, centered at (0,0).

0, 1, 1, 7, 13, 19, 31, 43, 55, 73, 85, 109, 121, 151, 169, 199, 235, 253, 295, 313, 361, 397, 433, 475, 511, 571, 595, 661, 703, 757, 817, 859, 925, 979, 1039, 1111, 1159, 1237, 1285, 1381, 1453, 1519, 1597, 1663, 1759, 1813, 1915, 1993, 2077, 2173, 2257

Offset: 0

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 13 2000

Keywords

• easy
• nonn

Comments

a(n)/(n/2)^2 --> Pi*2/sqrt(3) (see A186706).

Crossrefs

Cf. A053416 (closed disk), A053456, A053457, A053459.

Cf. A186706.

Programs

Maple

A053458 := 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 := (-j-sqrt(d^2-3*j^2))/2 ;
        if type(imin,integer) then
            imin := imin+1 ;
        else
            imin := ceil(imin) ;
        end if;
        imax := (-j+sqrt(d^2-3*j^2))/2 ;
        if type(imax,integer) then
            imax := imax -1 ;
        else
            imax := floor(imax) ;
        end if;
        a := a+imax-imin+1 ;
    end do:
    a ;
end proc:
seq(A053458(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, Sep 05 2023 *)