A346993 -id:A346993 - cp's OEIS frontend

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

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.

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

A192493 Numerators of squared radii of circumcircles of non-degenerate triangles with integer vertex coordinates.

Original entry on oeis.org

1, 1, 5, 25, 25, 2, 5, 25, 25, 13, 325, 169, 65, 4, 65, 17, 425, 221, 9, 289, 1105, 169, 85, 5, 325, 85, 50, 1105, 289, 25, 2125, 625, 13, 325, 425, 1625, 169, 1105, 125, 65, 29, 2465, 4225, 1885, 725, 377, 2465, 5525, 1885, 125, 8, 145, 65, 841, 17, 841, 845, 425, 2125, 221, 6409, 9425, 9, 325, 289, 145, 1105, 37, 5365, 3145, 169, 2405, 925, 85, 1369, 4625, 481, 625, 493, 2405, 10

Offset: 1

Hugo Pfoertner, Jul 10 2011

			The smallest triangle of lattice points {(0,0),(1,0),(0,1)} has circumradius R=sqrt(2)/2, i.e., R^2=1/2. Therefore a(1)=1, A192494(1)=2.

Hugo Pfoertner, Table of n, a(n) for n = 1..9089, covering range R^2 <= 100.
Hugo Pfoertner, Circles Passing through 3 Points of the Square Lattice, illustrations up to R^2=10.

Cf. A192494 (corresponding denominators), A128006, A128007.

Cf. A346993, A346994, A346995.

A192494 Denominators of squared radii of circumcircles of non-degenerate triangles with integer vertex coordinates.

Original entry on oeis.org

2, 1, 4, 18, 16, 1, 2, 9, 8, 4, 98, 50, 18, 1, 16, 4, 98, 50, 2, 64, 242, 36, 18, 1, 64, 16, 9, 196, 50, 4, 338, 98, 2, 49, 64, 242, 25, 162, 18, 9, 4, 338, 578, 256, 98, 50, 324, 722, 242, 16, 1, 18, 8, 100, 2, 98, 98, 49, 242, 25, 722, 1058, 1, 36, 32, 16, 121, 4, 578, 338, 18, 256, 98, 9, 144, 484, 50, 64, 50, 242, 1

Offset: 1

Hugo Pfoertner, Jul 10 2011

			The smallest triangle of lattice points {(0,0),(1,0),(0,1)} has circumradius R=sqrt(2)/2, i.e., R^2=1/2. Therefore A192493(1)=1, a(1)=2.

Hugo Pfoertner, Table of n, a(n) for n = 1..9089, covering range R^2 <= 100.

Cf. A192493 (corresponding numerators).

Cf. A346993, A346994, A346995.

A346124 Numbers m such that no self-avoiding walk of length m + 1 on the square lattice fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 4, 6, 8, 12, 14, 15, 16, 18, 20, 21, 23, 24, 25, 26, 27, 28, 32, 34, 36, 38, 44, 46, 48, 52, 56, 58, 60

Offset: 1

Hugo Pfoertner and Markus Sigg, Jul 30 2021

Closed walks are allowed.

			See link for illustrations of terms corresponding to diameters D < 8.5.

Hugo Pfoertner, Examples of paths of maximum length.

The squared radii of the enclosing circles are a subset of A192493/A192494.
Cf. A037245, A122224, A127399, A127400, A127401, A266549, A316194, A346993, A346994, A346995.
Cf. A346123-A346132 similar to this sequence with other sets of turning angles.

A356462 a(n) is the maximum number of Z x Z lattice points inside or on a circle of radius n^(1/2) for any position of the center of the circle.

Original entry on oeis.org

1, 5, 9, 12, 14, 21, 21, 24, 28, 32, 37, 37, 41, 45, 48, 52, 52, 57, 61, 63, 69, 69, 72, 76, 78, 81, 89, 89, 92, 97, 97, 100, 104, 112, 112, 115, 116, 121, 122, 127, 129, 137, 137, 140, 144, 148, 148, 152, 155, 157, 161, 164, 169, 177, 177

Offset: 0

Bernard Montaron, Aug 08 2022

nonn

a(n) >= A057655(n).

The terms of square index of this sequence are such that a(n^2) = A123690(2n), e.g., a(9) = 32 = A123690(6).

			For n = 1 the maximum number of Z x Z lattice points inside the circle is a(1) = 5. The maximum is obtained with the circle centered at x = 0, y = 0.

Cf. A057655, A123690, A346993.

Let N(u,v,n) be the number of integer solutions (x,y) of (x-u)^2 + (y-v)^2 <= n. Then a(n) is the maximum of N(u,v,n) taken over 0 <= u <= 1/2 and 0 <= v <= u. The symmetries of the square lattice allow to limit the domain of the circle center (u,v) to this triangle. The terms of this sequence were found by "brute force" search of the maximum of N(u,v,n) for (u,v) in this triangular domain.

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

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

Views

Author

Keywords

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A192493 Numerators of squared radii of circumcircles of non-degenerate triangles with integer vertex coordinates.

Views

Author

Keywords

Examples

Links

Crossrefs

A192494 Denominators of squared radii of circumcircles of non-degenerate triangles with integer vertex coordinates.

Views

Author

Keywords

Examples

Links

Crossrefs

A346124 Numbers m such that no self-avoiding walk of length m + 1 on the square lattice fits into the smallest circle that can enclose a walk of length m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A356462 a(n) is the maximum number of Z x Z lattice points inside or on a circle of radius n^(1/2) for any position of the center of the circle.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula