A125850 -id:A125850 - cp's OEIS frontend

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

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 10 2000

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

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

Cf. A003215, A125849, A125850, A125851, A125852.

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

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

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 *)

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

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A053479 Circle numbers (version 6): a(n) = number of points (i+j/2,jsqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (1/2, 1/(2sqrt(3))).

Original entry on oeis.org

0, 0, 3, 6, 12, 21, 30, 42, 54, 69, 90, 102, 129, 150, 174, 198, 225, 258, 288, 327, 354, 396, 435, 471, 522, 558, 609, 654, 702, 759, 807, 864, 924, 981, 1038, 1104, 1173, 1230, 1308, 1368, 1443, 1512, 1590, 1671, 1746, 1830, 1908, 2001, 2076, 2166, 2265

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 14 2000

In other words, number of points in a hexagonal lattice covered by a circular disk of diameter n if the center of the circle is chosen at the deep hole. - Hugo Pfoertner, Jan 07 2007

Also number of integer coordinate pairs (s,t) satisfying s^2+t^2+st-s-t <= n^2/4-1/3. The a(2)=3 coordinate pairs are (s,t)=(0,0), (0,1) and (1,0). The a(3)=6 coordinate pairs are (-1,1),(0,0),(0,1),(1,-1),(1,0) and (1,1). - R. J. Mathar, Feb 23 2007

Cf. A053411, A053414, A053415, A053416, A053417.

Cf. A005882, A125850, A125851, A125852.

A053479 := proc(n) local res,a,b ; res :=0 ; for a from -n to n do for b from -n to n do if a^2+b^2+a*b-a-b <= n^2/4-1/3 then res := res+1 ; fi ; od ; od ; RETURN(res) ; end : for n from 1 to 40 do printf("%d ",A053479(n)) ; od ; # R. J. Mathar, Feb 23 2007

cx = 1/2; cy = 1/(2*Sqrt[3]); a[n_] := Sum[ dj = (1/2)* Sqrt[Abs[-3*cx^2 + 2*Sqrt[3]*cx*cy - cy^2 + 6*cx*i - 2*Sqrt[3]*cy*i - 3*i^2 + n^2]]; j1 = cx/2 + (Sqrt[3]*cy)/2 - i/2 - dj // Floor ; j2 = cx/2 + (Sqrt[3]*cy)/2 - i/2 + dj // Ceiling; Sum[Boole[(i + j/2 - cx)^2 + (j*(Sqrt[3]/2) - cy)^2 <= n^2/4], {j, j1, j2}], {i, -(n + 1)/2 - 2 // Floor, (n + 1)/2 + 3 // Ceiling}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 06 2013 *)

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

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A053479 Circle numbers (version 6): a(n) = number of points (i+j/2,jsqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (1/2, 1/(2sqrt(3))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A053479 Circle numbers (version 6): 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 (1/2, 1/(2*sqrt(3))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A053479 Circle numbers (version 6): a(n) = number of points (i+j/2,jsqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (1/2, 1/(2sqrt(3))).