A053414 -id:A053414 - 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

A053417 Circle numbers (version 5): 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 (1/2,0).

Original entry on oeis.org

0, 2, 4, 10, 14, 24, 30, 48, 60, 76, 92, 110, 130, 154, 178, 208, 230, 264, 288, 330, 364, 406, 442, 482, 522, 564, 614, 664, 712, 766, 812, 874, 922, 990, 1050, 1112, 1176, 1240, 1312, 1382, 1452, 1530, 1598, 1684, 1750, 1840, 1920, 2008, 2092, 2182, 2266

Offset: 0

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

Equivalently, 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 middle between two lattice points. - Hugo Pfoertner, Jan 07 2007

Same as above but "number of disks (r = 1)" instead of "number of points". a(2^n - 1) = A239073(n), n >= 1. 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. A053411, A053414, A053415, A053416, A053417.
Cf. A053479, A125851, A125852.
Cf. A239073.

a[n_] := Sum[dj = Sqrt[Abs[4*n^2 + 6*i - 3*i^2 - 3]]/4; j1 = (1 - 2*i)/4 - dj // Floor; j2 = (1 - 2*i)/4 + dj // Ceiling; Sum[ Boole[i^2 - i - j/2 + i*j + j^2 + 1/4 <= n^2/4], {j, j1, j2}], {i, -n - 1, n + 3}]; 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

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

A123690 Number of points in a square lattice covered by a circle of diameter n if the center of the circle is chosen such that the circle covers the maximum possible number of lattice points.

Original entry on oeis.org

2, 5, 9, 14, 22, 32, 41, 52, 69, 81, 97, 116, 137, 157, 180, 208, 231, 258, 293, 319, 351, 384, 421, 457, 495, 540, 578, 623, 667, 716, 761, 812, 861, 914, 973, 1025, 1085, 1142, 1201, 1268, 1328, 1396, 1460, 1528, 1597, 1669, 1745, 1816, 1893, 1976, 2053

Offset: 1

Hugo Pfoertner, Oct 09 2006, Feb 11 2007

nonn

a(n) >= max(A053411(n), A053414(n), A053415(n)).

a(n) is an upper bound for the number of segments of a self avoiding path on the 2-dimensional square lattice such that the path fits into a circle of diameter n. A122224(n) <= a(n).

			a(1)=2: Circle with diameter 1 and center (0,0.5) covers 2 lattice points;
a(2)=5: Circle with diameter 2 and center (0,0) covers 5 lattice points;
a(3)=4: Circle with diameter 3 and center (0,0) covers 9 lattice points;
a(4)=14: Circle with diameter 4 and center (0.5,0.2) covers 14 lattice points.

Hugo Pfoertner, Maximum number of points in the square lattice covered by circular disks. Illustrations.

Cf. A123689, A053411, A053414, A053415, A122224, A295344, A291259, A346993, A346994, A346995.

The corresponding sequences for the hexagonal lattice and the honeycomb net are A125852 and A127406, respectively.

(* An exact program using the functions from A291259: *)
Clear[a]; a[n_] := Module[{points, pairc, expcent, innerpoints, cn=Ceiling[n], allpairs},
allpairs = Flatten[Table[{i, j}, {i, -cn, cn+1}, {j, -cn, cn+1}], 1];
points = Select[allpairs, candidatePointQ[#, n]&];
pairc = Select[Subsets[points, {2}], dd2@@#<=4n^2&];
expcent = explorativeCenters[pairc, n];
innerpoints = Count[allpairs, _?(innerPointQ[#, n]&)];
Max[Table[Count[points, _?(dd2[#, center]<=n^2&)], {center, expcent}]] + innerpoints];
Table[a[n/2], {n, 20}] (* Andrey Zabolotskiy, Feb 21 2018 *)

a(21)-a(40) originally conjectured by Jean-François Alcover confirmed and moved to Data and more terms added by Andrey Zabolotskiy, Feb 21 2018

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 *)

A346993 Record numbers of grid points in a square lattice covered by a continuously growing circular disk if the center of the disk is chosen to cover the maximum possible number of grid points.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 9, 12, 13, 14, 16, 17, 21, 22, 24, 26, 27, 28, 32, 33, 37, 38, 39, 40, 41, 44, 45, 46, 47, 48, 52, 56, 57, 58, 59, 61, 62, 63, 64, 65, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 89, 90, 91, 92, 93, 94, 97, 98, 99, 100, 104, 112, 113

Offset: 1

Hugo Pfoertner, Aug 16 2021

nonn

			     Diameter  Covered      R^2 =
     of disk   grid        (D/2)^2 =
   n    D      points  A346994(n)/A346995(n)
.
   1 0.00000     1           0   /    1
   2 1.00000     2           1   /    4
   3 1.41421     4           1   /    2
   4 2.00000     5           1   /    1
   5 2.23607     6           5   /    4
   6 2.50000     7          25   /   16
   7 2.82843     9           2   /    1
   8 3.16228    12           5   /    2
   9 3.67696    13         169   /   50
  10 3.80058    14          65   /   18
  11 4.12311    16          17   /    4
  12 4.33333    17         169   /   36
  13 4.47214    21           5   /    1

Hugo Pfoertner, Visualization of configurations with maximum number of covered grid points.

The corresponding squared radii are A346994/A346995.

Cf. A053411, A053414, A053415, A123690, A346124.

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

A123689 Number of points in a square lattice covered by a circle of diameter n if the center of the circle is chosen such that the circle covers the minimum possible number of lattice points.

Original entry on oeis.org

0, 2, 4, 10, 16, 26, 32, 46, 60, 74, 88, 108, 124, 146, 172, 194, 216, 248, 276, 308

Offset: 1

Hugo Pfoertner, Oct 09 2006

a(n)<=min(A053411(n),A053414(n),A053415(n)).

Using brute force computation and a step size of 1/1000 (though 1/200 suffices), the [conjectured] terms a(21) to a(40) would be: 332, 374, 408, 438, 484, 522, 560, 608, 648, 698, 740, 794, 846, 894, 952, 1006, 1060, 1124, 1184, 1248. - Jean-François Alcover, Jan 08 2018

			a(1)=0: Circle with diameter 1 with center (0.5,0.5) covers no lattice points; a(2)=2: Circle with diameter 2 with center (0,eps) covers 2 lattice points;
a(3)=4: Circle with diameter 3 with center (0.5,0.5) covers 4 lattice points.

Hugo Pfoertner, Minimal number of points in the square lattice covered by circular disks. Illustrations.

Cf. A123690, A053411, A053414, A053415, A122224, A291259.

The corresponding sequences for the hexagonal lattice and the honeycomb net are A125851 and A127405, respectively.

dx = 1/200; y0 = 0; (* To speed up computation, the step size dx is experimentally adjusted and the circle center is taken on the x-axis. *)
cnt[pts_, ctr_, r_] := Count[pts, pt_ /; Norm[pt - ctr] <= r];
a[n_] := Module[{r, pts, innerCnt, an, center}, r = n/2; pts = Select[ Flatten[ Table[{x, y}, {x, -r - 1, r + 1}, {y, -r - 1, r + 1}], 1], r - 1 <= Norm[#] <= r + 1 &]; innerCnt = Sum[If[Norm[{x, y}] < r - 1, 1, 0], {x, -r - 1, r + 1}, {y, -r - 1, r + 1}]; {an, center} = Table[{innerCnt + cnt[pts, {x, y0}, r], {x, y0}}, {x, -1/2, 1/2, dx}] // Sort // First; Print["a(", n, ") = ", an, ", center = ", center // InputForm]; an];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Jan 08 2018 *)

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.

A053457 Open disk numbers (version 2): a(n) is the number of points (i,j), i,j, integers, contained in an open disk of diameter n, centered at (0,1/2).

Original entry on oeis.org

0, 0, 2, 6, 12, 16, 26, 38, 48, 60, 78, 94, 108, 128, 154, 174, 196, 224, 250, 278, 312, 344, 378, 410, 452, 484, 526, 574, 612, 652, 698, 754, 800, 848, 906, 958, 1012, 1068, 1130, 1190, 1252, 1316, 1378, 1446, 1516, 1588, 1654, 1730, 1808, 1880, 1954

Offset: 0

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

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

Cf. A053414, A053456, A053458, A053459.

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).

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A053417 Circle numbers (version 5): 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 (1/2,0).

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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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A123690 Number of points in a square lattice covered by a circle of diameter n if the center of the circle is chosen such that the circle covers the maximum possible number of lattice points.

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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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A346993 Record numbers of grid points in a square lattice covered by a continuously growing circular disk if the center of the disk is chosen to cover the maximum possible number of grid 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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A123689 Number of points in a square lattice covered by a circle of diameter n if the center of the circle is chosen such that the circle 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,jsqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (1/2, 1/(2sqrt(3))).