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
Programs
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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
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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 *)
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