A295344 -id:A295344 - cp's OEIS frontend

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.

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

A291259 Minimum number of points of the square lattice falling strictly inside a circle of radius n.

Original entry on oeis.org

0, 1, 9, 25, 45, 69, 108, 145, 193, 248, 305, 373, 437, 517, 608, 697, 793, 889, 1005, 1124, 1245, 1369, 1510, 1649, 1789, 1941, 2109, 2278, 2449, 2617, 2809, 2997, 3202, 3405, 3613, 3834, 4049, 4281, 4509, 4762, 5013, 5249, 5521, 5785, 6068, 6348, 6621, 6917

Offset: 0

Andres Cicuttin, Aug 21 2017

nonn

Due to the symmetry and periodicity of the square lattice it is sufficient to explore possible circles with center belonging to the triangle with vertices (0,0), (1/2,0), and (1/2,1/2).

The different regions for the centers producing constant numbers of lattice points inside circles of radius n seem to become very complex and irregular as n increases (see density plots in Links).

			From _Arkadiusz Wesolowski_, Dec 18 2017 [Corrected by _Andrey Zabolotskiy_, Feb 19 2018]: (Start)
For a circle centered at the point (x, y) = (1/2, 0) with radius 6, there are 108 lattice points inside the circle.
Possible (but not unique) choices for the centers of the circles for radii up to 20 are given below.
.
.  Poss. center          Points in
.    x      y    Radius  the circle
.  -----  -----  ------  ----------
.    0      0       1         1
.    0      0       2         9
.    0      0       3        25
.    0      0       4        45
.    0      0       5        69
.   1/2     0       6       108
.    0      0       7       145
.    0      0       8       193
.   1/5     0       9       248
.    0      0      10       305
.    0      0      11       373
.    0      0      12       437
.    0      0      13       517
.   1/4     0      14       608
.    0      0      15       697
.    0      0      16       793
.    0      0      17       889
.    0      0      18      1005
.   1/2    1/2     19      1124
.    0      0      20      1245
(End)

Robert G. Wilson v, Table of n, a(n) for n = 0..125
Andres Cicuttin, Plots of regions for centers of circles of several radii n enclosing constant number of lattice points
Eric Weisstein's World of Mathematics, Circle Lattice Points

Cf. A046109, A051132, A295344.

(* A291259: Minimum number of points of the square lattice falling strictly inside a circle of radius n. *)
(* The three vertices of the Explorative Triangle (ET) *)
P1={0,0}; P2={1/2,0}; P3={1/2,1/2};
dd2=SquaredEuclideanDistance;
(* candidatePointQ[p,n] gives True if "p" is a candidate point, and False otherwise. A candidate point is a point belonging to a circle of radius "n" with center in the ET *)
candidatePointQ[p_,n_] := With[{dds={dd2[p,P1],dd2[p,P2],dd2[p,P3]}}, Max[dds]>=n^2>=Min[dds]];
(* Check if point "p" falls inside any circle with radius "n" and center in the ET *)
innerPointQ[p_,n_] := With[{dds={dd2[p,P1],dd2[p,P2],dd2[p,P3]}}, Max[dds]Andres Cicuttin & Andrey Zabolotskiy, Nov 14 2017 *)

a(n) ~ Pi*n^2.

a(n) <= A051132(n). - Joerg Arndt, Oct 03 2017

More terms from Andrey Zabolotskiy, Nov 17 2017

A355760 a(n) is the number of grid points in a square lattice covered by the area enclosed by n loops of an Archimedean spiral with starting point (0, 0) and endpoint (n, 0).

Original entry on oeis.org

1, 2, 8, 21, 40, 64, 97, 132, 178, 228, 282, 350, 415, 492, 574, 660, 756, 855, 962, 1076, 1195, 1322, 1451, 1590, 1736, 1885, 2044, 2204, 2378, 2552, 2734, 2922, 3116, 3317, 3525, 3741, 3960, 4187, 4416, 4655, 4900, 5154, 5410, 5674, 5946, 6223, 6502, 6791, 7087, 7391, 7698

Offset: 0

Karl-Heinz Hofmann, Jul 16 2022

nonn

Grid points coincident with the outer boundary of the spiral are included.
The spiral figure is closed with a line from (n, 0) to (n-1, 0).
Conjecture: Only lattice points on the positive x-axis are on the outer boundary of the spiral. It seems that the spiral passes all other grid points without hitting any.

			See the PDF in links.

Karl-Heinz Hofmann, Examples of a(1..8)
Wikipedia, Archimedean spiral

Cf. A000328, A295344, A046109.

cp's OEIS Frontend

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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A291259 Minimum number of points of the square lattice falling strictly inside a circle of radius n.

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A355760 a(n) is the number of grid points in a square lattice covered by the area enclosed by n loops of an Archimedean spiral with starting point (0, 0) and endpoint (n, 0).

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