A346785 -id:A346785 - 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.

A346784 Numerators of minimal squared radii of circular disks covering a record number of lattice points of the hexagonal lattice, when the centers of the circles are chosen to maximize the number of covered lattice points.

Original entry on oeis.org

0, 1, 1, 3, 1, 7, 49, 9, 7, 13, 169, 91, 4, 133, 21, 361, 1729, 169, 19, 7, 961, 133, 9, 39, 21793, 481, 31, 9331, 301, 3367, 49, 817, 13, 361, 931, 1813, 63, 16

Offset: 1

Hugo Pfoertner, Aug 08 2021

It is conjectured that the number of covered grid points is given by A346126(n-1) for n>2.

			0, 1/4, 1/3, 3/4, 1, 7/4, 49/25, 9/4, 7/3, 13/4, 169/48, 91/25, 4, 133/27, 21/4, 361/64, 1729/289, 169/27, 19/3, 7, 961/121, 133/16, 9, 39/4, 21793/2187, ...
.
     Diameter  Covered      R^2 =
     of disk   grid        (D/2)^2 =
   n    D      points    a(n) / A346785(n)
.
   1 0.00000     1        0   /    1
   2 1.00000     2        1   /    4
   3 1.15470     3        1   /    3
   4 1.73205     4        3   /    4
   5 2.00000     7        1   /    1
   6 2.64575     8        7   /    4
   7 2.80000     9       49   /   25
   8 3.00000    10        9   /    4
   9 3.05505    12        7   /    3
  10 3.60555    14       13   /    4

Hugo Pfoertner, Examples of self avoiding walks of minimum diameter on the hexagonal lattice.

Corresponding denominators are A346785.

Cf. A125852, A346126.

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.

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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.

Views

Author

Keywords

Examples

Links

Crossrefs

A346784 Numerators of minimal squared radii of circular disks covering a record number of lattice points of the hexagonal lattice, when the centers of the circles are chosen to maximize the number of covered lattice points.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs