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

0,4

Comments

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

Links

Table of n, a(n) for n=0..50.

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 *)

Crossrefs

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

Cf. A186706.

Sequence in context: A091180 A272381 A242190 * A040058 A172057 A173176

Adjacent sequences: A053455 A053456 A053457 * A053459 A053460 A053461

Keyword

easy,nonn

Author

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

Status

approved