A123690 -id:A123690 - cp's OEIS frontend

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.

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

A122224 Length of the longest possible self avoiding path on the 2-dimensional square lattice such that the path fits into a circle of diameter n.

Original entry on oeis.org

1, 4, 8, 14, 21, 32, 40

Offset: 2

Hugo Pfoertner, Sep 25 2006

The path may be open or closed. For larger n several solutions with the same number of segments exist.

It is conjectured that a(n) >= A123690(n)-1, i.e., that it is always possible to find a path visiting all grid points covered by a circle, irrespective of the position of its center. - Hugo Pfoertner, Mar 02 2018

Hugo Pfoertner, Examples of compact self avoiding paths on a square lattice.

Cf. A122223, A122226, A123690.

A127406 Number of points in a 2-dimensional honeycomb net covered by a circular disk of diameter n if the center of the disk is chosen to maximize the number of net points covered by the disk.

Original entry on oeis.org

2, 6, 7, 13, 17, 25, 34, 42, 54, 64, 78, 90, 107, 126, 140, 163, 178, 204, 222, 246

Offset: 1

Hugo Pfoertner, Feb 08 2007

a(n)>= max(A127402(n),A127403(n),A127404(n)). a(n) is an upper limit for the number of path segments in A122223.

Hugo Pfoertner, Maximal number of points in the honeycomb lattice covered by circular disks. Illustrations.

Cf. A127402, A127403, A127404, A127405, A122223. The corresponding sequences for the square lattice and hexagonal lattice are A123690 and A125852, respectively.

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.

A354704 T(w,h) is a lower bound for the maximum number of grid points in a square grid covered by an arbitrarily positioned and rotated rectangle of width w and height h, excluding the trivial case of an axis-parallel unshifted cover, where T(w,h) is a triangle read by rows.

Original entry on oeis.org

2, 3, 5, 5, 8, 13, 6, 10, 15, 18, 8, 12, 20, 24, 32, 9, 14, 23, 27, 36, 41, 10, 17, 25, 30, 40, 45, 53, 12, 19, 30, 36, 48, 54, 60, 72, 13, 21, 33, 39, 52, 59, 68, 78, 89, 15, 23, 38, 45, 60, 68, 75, 90, 98, 113, 16, 25, 40, 48, 64, 72, 81, 96, 105, 120, 128, 17, 28, 43, 52, 68, 77, 88, 102, 114, 128, 137, 149

Offset: 1

Hugo Pfoertner, Jun 15 2022

Grid points must lie strictly within the covering rectangle, i.e., grid points on the perimeter of the rectangle are not allowed. See A354702 for more information.

			The triangle begins:
    \ h  1   2   3   4   5   6   7    8    9   10   11   12
   w \ ----------------------------------------------------
   1 |   2;  |   |   |   |   |   |    |    |    |    |    |
   2 |   3,  5;  |   |   |   |   |    |    |    |    |    |
   3 |   5,  8, 13;  |   |   |   |    |    |    |    |    |
   4 |   6, 10, 15, 18;  |   |   |    |    |    |    |    |
   5 |   8, 12, 20, 24, 32;  |   |    |    |    |    |    |
   6 |   9, 14, 23, 27, 36, 41;  |    |    |    |    |    |
   7 |  10, 17, 25, 30, 40, 45, 53;   |    |    |    |    |
   8 |  12, 19, 30, 36, 48, 54, 60,  72;   |    |    |    |
   9 |  13, 21, 33, 39, 52, 59, 68,  78,  89;   |    |    |
  10 |  15, 23, 38, 45, 60, 68, 75,  90,  98, 113;   |    |
  11 |  16, 25, 40, 48, 64, 72, 81,  96, 105, 120, 128;   |
  12 |  17, 28, 43, 52, 68, 77, 88, 102, 114, 128, 137, 149

Hugo Pfoertner, Table of n, a(n) for n = 1..210, rows 1..20 of triangle, flattened
Hugo Pfoertner, Illustrations of the initial terms up to T(5,5).

Cf. A293330, A354702, A354705, A354706 (diagonal), A354707, A355244.

Cf. A123690 (similar problem with circular disks).

PARI
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\\ See links in A354702 and A355244.
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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.

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 *)

A346994 Numerators of the squared radii corresponding to circular disks covering record numbers of grid points A346993 of the square lattice.

Original entry on oeis.org

0, 1, 1, 1, 5, 25, 2, 5, 169, 65, 17, 169, 5, 25, 13, 29, 5525, 125, 17, 481, 10, 45, 205, 19721, 1189, 25, 13, 338, 29725, 697, 29, 65, 17, 1105, 3445, 18, 4453, 40885, 4625, 481, 20, 85, 12505, 2125, 200, 89, 45, 7921, 425, 89725, 93925, 2405, 2465, 10201, 98345

Offset: 1

Hugo Pfoertner, Aug 16 2021

			0/1, 1/4, 1/2, 1/1, 5/4, 25/16, 2/1, 5/2, 169/50, 65/18, 17/4, 169/36, ...
For detailed examples see A346993 and the linked pdf.

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

The corresponding denominators are A346995.
Cf. A346993, A123690, A346124, A346784, A346785.
All terms of a(n)/A346995(n) with the sole exception of 1/4 are terms of A192493/A192494.

A346995 Denominators of the squared radii corresponding to circular disks covering record numbers of grid points A346993 of the square lattice.

Original entry on oeis.org

1, 4, 2, 1, 4, 16, 1, 2, 50, 18, 4, 36, 1, 4, 2, 4, 722, 16, 2, 50, 1, 4, 18, 1682, 100, 2, 1, 25, 2178, 50, 2, 4, 1, 64, 196, 1, 242, 2178, 242, 25, 1, 4, 578, 98, 9, 4, 2, 338, 18, 3698, 3844, 98, 98, 400, 3844

Offset: 1

Hugo Pfoertner, Aug 16 2021

			0/1, 1/4, 1/2, 1/1, 5/4, 25/16, 2/1, 5/2, 169/50, 65/18, 17/4, 169/36, ...
For detailed examples see A346993 and the linked pdf.

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

The corresponding numerators are A346994.

Cf. A346993, A123690, A346124, A346784, A346785.

A295344 Maximum number of lattice points inside and on a circle of radius n.

Original entry on oeis.org

1, 5, 14, 32, 52, 81, 116, 157, 208, 258, 319, 384, 457, 540, 623, 716, 812, 914, 1025, 1142, 1268, 1396, 1528, 1669, 1816, 1976, 2131, 2300, 2472, 2650, 2836, 3028, 3228, 3436, 3644, 3859, 4080, 4314, 4548, 4792, 5038, 5289, 5555, 5818, 6092, 6376, 6668, 6952

Offset: 0

Arkadiusz Wesolowski, Nov 20 2017

nonn

Maximum number of lattice points (i.e., points with integer coordinates) in the plane that can be covered by a circle of radius n.
a(n) >= A000328(n).
Conjecture: sequence contains infinitely many terms that are divisible by 4.

			For a circle centered at the point (x, y) = (1/2, 1/4) with radius 2, there are 14 lattice points inside and on the circle.
.
.     Center             # Pts in/
.    x      y    Radius  on circle
.  -----  -----  ------  ---------
.    0      0       1         5
.   1/2    1/4      2        14
.   1/2    1/2      3        32
.   1/2    1/2      4        52
.    0      0       5        81
.   1/2    1/3      6       116
.   2/5    1/5      7       157
.   1/2    1/2      8       208
.   1/2    2/9      9       258
.  20/47  19/56    10       319
.   1/2    1/2     11       384
.  11/23   7/20    12       457
.   1/2    1/2     13       540
.  10/21   3/13    14       623
.   1/2    1/2     15       716
.   1/2    1/2     16       812
.   2/5    2/5     17       914
.   3/8    5/14    18      1025
.   1/2    1/6     19      1142
.   9/19   8/17    20      1268

B. R. Srinivasan, Lattice Points in a Circle, Proc. Nat. Inst. Sci. India, Part A, 29 (1963), pp. 332-346.

Cf. A000328, A123690, A291259.

L=List([]); for(n=0, 47, if(n>0, j=5, j=1); g=0; h=0; f=ceil(Pi*n^2); for(d=2, floor(f/2), for(c=1, floor(d/2), if(gcd(c, d)==1, for(e=d, d+1, if(e/f<=1/2, a=c/d; b=e/f; if(a+b>=1/2, t=0; for(x=-n, n+1, for(y=-n, n+1, z=(a-x)^2+(b-y)^2; if(z<=n^2,t++))); if(t>j, j=t; if(a>=b, g=a; h=b, g=b; h=a)))))))); print("a("n") = "j", the center of the circle is at point ("g", "h")."); listput(L, j)); print(); print(Vec(L));

a(n) = Pi*n^2 + O(n), as n goes to infinity.

a(n) = A123690(2*n) for n >= 1.

a(10) corrected by Giovanni Resta, Nov 24 2017

A386538 a(n) is the maximum possible area of a polygon within a circle of radius n, where both the center and the vertices lie on points of a unit square grid.

Original entry on oeis.org

0, 2, 8, 24, 42, 74, 104, 138, 186, 240, 304, 362, 424, 512, 594, 690, 776, 880, 986, 1104, 1232, 1346, 1490, 1624, 1762, 1930, 2088, 2256, 2418, 2594, 2784, 2962, 3170, 3368, 3584, 3810, 4008, 4248, 4466, 4730, 4976, 5210, 5474, 5736, 6024, 6306, 6570, 6864, 7154

Offset: 0

Felix Huber, Aug 05 2025

nonn

a(n) > 99% of the circle area for n >= 50.
Conjecture: The maximum possible area of a polygon within the circle would be the same if only the vertices but not the center were fixed on grid points.
All terms are even.

			See linked illustration of the term a(4) = 42.

Felix Huber, Table of n, a(n) for n = 0..10000
Felix Huber, Illustration of a(4) = 42
Eric Weisstein's World of Mathematics, Pick's Theorem

Cf. A000217, A066643, A123690, A125228, A288247, A322106, A322107, A357577, A386539.

A386538:=proc(n)
    local x,y,p,s;
    p:=4*n;
    s:={};
    for x to n do
        y:=floor(sqrt(n^2-x^2));
        p:=p+4*y;
        s:=s union {y}
    od;
    return p-2*nops(s)
end proc;
seq(A386538(n),n=0..48);

a[n_] := Module[{p=4n},s = {}; Do[ y = Floor[Sqrt[n^2 - x^2]];p = p + 4*y;s = Union[s, {y}],{x,n} ];p - 2*Length[s]];Array[a,49,0] (* James C. McMahon, Aug 19 2025 *)

a(n) = A386539(A000217(n)) = A386539(n,n) for n >= 1.

a(n) <= A066643(n).

cp's OEIS Frontend

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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A122224 Length of the longest possible self avoiding path on the 2-dimensional square lattice such that the path fits into a circle of diameter n.

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A127406 Number of points in a 2-dimensional honeycomb net covered by a circular disk of diameter n if the center of the disk is chosen to maximize the number of net points covered by the disk.

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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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A354704 T(w,h) is a lower bound for the maximum number of grid points in a square grid covered by an arbitrarily positioned and rotated rectangle of width w and height h, excluding the trivial case of an axis-parallel unshifted cover, where T(w,h) is a triangle read by rows.

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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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A346994 Numerators of the squared radii corresponding to circular disks covering record numbers of grid points A346993 of the square lattice.

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A346995 Denominators of the squared radii corresponding to circular disks covering record numbers of grid points A346993 of the square lattice.

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A295344 Maximum number of lattice points inside and on a circle of radius n.

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A386538 a(n) is the maximum possible area of a polygon within a circle of radius n, where both the center and the vertices lie on points of a unit square grid.

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