A053417 -id:A053417 - cp's OEIS frontend

A160826 Improvement of A125852 over A053416, A053479 and A053417.

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 4, 3, 0, 0, 0, 1, 0, 0, 0, 0, 2, 4, 5, 1, 3, 1, 0, 3, 2, 3, 4, 3, 4, 5, 6, 9, 4, 3, 0, 1, 0, 0, 0, 2, 4, 3, 4, 5, 10, 14, 3, 6, 0, 7, 0, 4, 5, 1, 8, 6, 0, 4, 7, 8, 6, 5, 11, 5, 9, 12, 12, 4, 0, 11, 7, 12, 0, 3, 1, 0, 1, 5, 0, 6, 2, 10, 11, 25, 17, 3, 2, 0, 9, 0, 12, 5, 0, 4, 2

Offset: 1

Hagen von Eitzen, May 27 2009

nonn

How many more lattice points of a hexagonal lattice can be covered by placing a disk of diameter n at an optimal center instead of one of the three obvious centers (a lattice point, midpoint between two lattice points, barycenter of a fundamental triangle)?
The first difference occurs at n=9, when a diameter 9 disc around e.g. (1/2, 4*sqrt(5)) covers more lattice points than one around (0,0) or (1/2,0) or (1/2,sqrt(3)/6).
Clearly a(n) = O(n) as all "extra" points have norm approximately n^2/4 if the optimal center is chosen near (0,0). Does a(n)/n converge? Are there only finitely many n with a(n)=0?

			For diameters n=2,4,6,8 a disc around (0,0) and for n=1,3,5,7 a disc around(1/2,0) happens to be optimal (covers as many points as possible); therefore a(1)=a(2)=...=a(8)=0.
a(9) = A125852(9) - max(A053416(9),A053479(9),A053417(9)) = 77 - max(73,69,76) = 1.

H. v. Eitzen, Table of n, a(n) for n=1..1000

a(n) = A125852(n) - max(A053416(n),A053479(n),A053417(n))

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).

Original entry on oeis.org

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

A125852 Number of points in a hexagonal lattice covered by a circular disk of diameter n if the center of the circle is chosen such that the disk covers the maximum possible number of lattice points.

Original entry on oeis.org

2, 7, 10, 19, 24, 37, 48, 61, 77, 94, 115, 134, 157, 187, 208, 241, 265, 301, 330, 367, 406, 444, 486, 527, 572, 617, 665, 721, 769, 825, 877, 935, 993, 1054, 1117, 1182, 1249, 1316, 1385, 1459, 1531, 1615, 1684, 1765, 1842, 1925, 2011, 2096, 2187, 2276

Offset: 1

Hugo Pfoertner, Jan 07 2007, Feb 11 2007

nonn

a(n)>=max(A053416(n),A053479(n),A053417(n)). a(n) is an upper bound for the number of segments of a self avoiding path on the 2-dimensional triangular lattice such that the path fits into a circle of diameter n. A122226(n)<=a(n).

H. v. Eitzen, Table of n, a(n) for n=1..1000
Index entries for sequences related to A2 = hexagonal = triangular lattice
Hugo Pfoertner, Maximum number of points in the hexagonal lattice covered by circular disks. Illustrations.

Cf. A053416, A053479, A053417, A125851, A122226. The corresponding sequences for the square lattice and the honeycomb net are A123690 and A127406, respectively.

More terms copied from b-file by Hagen von Eitzen, Jun 17 2009

A053415 Circle numbers (version 3): a(n) = number of points (i,j), i,j integers, contained in a circle of diameter n, centered at (1/2, 1/2).

Original entry on oeis.org

0, 0, 4, 4, 12, 16, 32, 32, 52, 60, 80, 88, 112, 124, 156, 172, 208, 216, 256, 276, 316, 332, 384, 408, 448, 484, 540, 560, 616, 648, 716, 740, 812, 848, 912, 952, 1020, 1060, 1124, 1184, 1264, 1304, 1396, 1436, 1528, 1576, 1664, 1716, 1804, 1876, 1976

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

cx = 1/2; cy = 1/2; a[n_] := Sum[ dj = 1/2*Sqrt[ Abs[n^2 - 4*cx^2 + 8*cx*i - 4*i^2]]; j1 = cy - dj // Floor; j2 = cy + dj // Ceiling; Sum[Boole[ (i - cx)^2 + (j - cy)^2 <= n^2/4], {j, j1, j2}], {i, cx - n/2 // Floor, cx + n/2 // Ceiling}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 06 2013 *)

a(n)/(n/2)^2 -> Pi.

a(n) = [x^(n^2)] theta_2(x^4)^2 / (1 - x). - Ilya Gutkovskiy, Nov 23 2021

A053411 Circle numbers (version 1): a(n)= number of points (i,j), i,j integers, contained in a circle of diameter n, centered at the origin.

Original entry on oeis.org

1, 1, 5, 9, 13, 21, 29, 37, 49, 69, 81, 97, 113, 137, 149, 177, 197, 225, 253, 293, 317, 349, 377, 421, 441, 489, 529, 577, 613, 665, 709, 749, 797, 861, 901, 973, 1009, 1085, 1129, 1201, 1257, 1313, 1373, 1457, 1517, 1597, 1653, 1741, 1793, 1885, 1961

Offset: 0

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

a(n)/(n/2)^2 -> Pi.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A053414, A053415, A053416, A053417.

Bisections: A000328 and A036704.

a[n_] := (m = Ceiling[n/2]; Sum[Boole[i^2 + j^2 <= n^2/4], {i, -m, m}, {j, -Ceiling @ Sqrt[ m^2 - i^2 ], Ceiling @ Sqrt[ m^2 - i^2 ]}]); Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 06 2013 *)

A053414 Circle numbers (version 2): a(n) is the number of points (i,j), i,j integers, contained in a circle of diameter n, centered at (0, 1/2).

Original entry on oeis.org

0, 2, 2, 8, 12, 22, 26, 40, 48, 62, 78, 96, 108, 134, 154, 180, 196, 230, 250, 280, 312, 346, 378, 412, 452, 494, 526, 576, 612, 658, 698, 756, 800, 850, 906, 964, 1012, 1074, 1130, 1196, 1252, 1322, 1378, 1448, 1516, 1594, 1654, 1732, 1808, 1882, 1954

Offset: 0

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

Lim_{n->infinity} a(n)/n^2 = Pi/4.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A053411, A053415, A053416, A053417.

a[n_] := (m = Ceiling[n/2]; Sum[ Boole[ n > 2*i && 2*i+n > 0 && n^2 - 4*i^2 + 4*j - 4*j^2 - 1 >= 0 ], {i, -m, m}, {j, -Ceiling @ Sqrt[ m^2-i^2 ], Ceiling @ Sqrt[ m^2-i^2 ]}]); Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 06 2013 *)

A125851 Number of points in a hexagonal lattice covered by a circular disk of diameter n if the center of the circle is chosen such that the disk covers the minimum possible number of lattice points.

Original entry on oeis.org

0, 3, 6, 12, 19, 30, 40, 54, 69, 87, 102, 123, 149, 174, 198, 225, 253, 287, 313, 354, 396, 435

Offset: 1

Hugo Pfoertner, Jan 07 2007, Feb 11 2007

a(n)<=min(A053416(n),A053479(n),A053417(n))

Index entries for sequences related to A2 = hexagonal = triangular lattice
Hugo Pfoertner, Minimal number of points in the hexagonal lattice covered by circular disks. Illustrations.

Cf. A053416, A053479, A053417, A125852. The corresponding sequences for the square lattice and the honeycomb net are A123689 and A127405, respectively.

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

A127404 Number of points in a honeycomb net covered by a circular disk of diameter n if the center of the circle is chosen at mid-edge between two grid points.

Original entry on oeis.org

2, 2, 6, 10, 16, 20, 34, 36, 50, 58, 72, 86, 106, 116, 138, 154, 176, 190, 222, 234, 270, 292

Offset: 1

Hugo Pfoertner, Feb 08 2007

			a(1)=2 because a disk of diameter 1 centered at the middle of an edge covers the 2 net points bounding this edge.

Cf. A127402, A127403, A127405, A127406. The corresponding sequences for the square lattice and hexagonal lattice are A053414 and A053417, respectively.

A239073 Given a circle of radius R into which small circles of radius R/2^n are packed in a "hexagonal pattern" (see Comments), a(n) is the maximum number of small circles.

Original entry on oeis.org

1, 2, 10, 48, 208, 874, 3600, 14622, 58976, 236824, 949094, 3800126, 15207812, 60846154, 243414326, 973716670, 3894985588, 15580180122, 62321195992, 249285735518, 997144844044, 3988583179554, 15954340324098, 63817376508852, 255269536476262

Offset: 0

Kival Ngaokrajang, Mar 10 2014

nonn

The construction rule is: (1) Start with a unit circle (big circle). (2) Pack circles at radius 1/2^n (small circles) on the diameter line of the big circle. (3) Maintain hexagonal packing pattern of small circles on the rows above and below that of the previous step. The number of small circles in any row is limited so that the circumference of the last small circle does not cross the circumference of the big circle (but is allowed to contact it). (4) Repeat process to the top and bottom rows.
See illustration in links.
From Jon E. Schoenfield, Mar 23 2014: (Start)
There can never be exactly one small circle on the top row iMax (and, symmetrically, exactly one on the bottom row) that contacts the large circle, since the upper edge of that small circle would be at a radius of (iMax*sqrt(3)+1)/2^n from the center of the unit circle (which cannot equal 1 unless iMax=0, i.e., there are no rows above or below the center).
The number of small circles on the top row seems to be on the order of 2^(n/2). It seems nearly certain that, for n>0, there is never exactly one small circle on the top row. (End)
After Wolfdieter Lang's suggestions of an alternative definition, I found later that a(n) = A053417(2^n-1), n >= 1.

Kival Ngaokrajang, Illustration of initial terms
Kival Ngaokrajang, Small Basic program

Cf. A239074, A239206, A053417, A053416.

for n in [0..25] do
   x:=(2^n - 1)^2;
   c:=2^n; // for row i=0
   for i in [1..Isqrt(x div 3)] do
      t:=x-3*i*i;
      if IsOdd(i) then
         c:=c+2*(2*Floor(Sqrt(t)/2)+1);
      else
         c:=c+2*(2*Floor(Sqrt(t)/2+1/2));
      end if;
   end for;
   n, c;
end for; // Jon E. Schoenfield, Mar 17 2014

a(n) = 2^n + 2 * Sum_{i odd, <= iMax} (2*floor(sqrt(x - 3 * i^2) / 2) + 1) + 2 * Sum_{i even, 0 < i <= iMax} 2*floor(sqrt(x - 3 * i^2) / 2 + 1/2) where x = (2^n-1)^2 and iMax = floor(sqrt(x/3)). - Jon E. Schoenfield, Mar 17, Mar 23 2014

a(20) - a(24) and corrections to a(15), a(17), and a(18) from Jon E. Schoenfield, Mar 17 2014

cp's OEIS Frontend

A160826 Improvement of A125852 over A053416, A053479 and A053417.

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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).

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A125852 Number of points in a hexagonal lattice covered by a circular disk of diameter n if the center of the circle is chosen such that the disk covers the maximum possible number of lattice points.

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A053415 Circle numbers (version 3): a(n) = number of points (i,j), i,j integers, contained in a circle of diameter n, centered at (1/2, 1/2).

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A053411 Circle numbers (version 1): a(n)= number of points (i,j), i,j integers, contained in a circle of diameter n, centered at the origin.

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A053414 Circle numbers (version 2): a(n) is the number of points (i,j), i,j integers, contained in a circle of diameter n, centered at (0, 1/2).

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A125851 Number of points in a hexagonal lattice covered by a circular disk of diameter n if the center of the circle is chosen such that the disk covers the minimum possible number of lattice points.

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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))).

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A127404 Number of points in a honeycomb net covered by a circular disk of diameter n if the center of the circle is chosen at mid-edge between two grid points.

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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))).