A127403 -id:A127403 - cp's OEIS frontend

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.

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.

A127405 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 minimize the number of net points covered by the disk.

Original entry on oeis.org

0, 2, 4, 8, 12, 20, 24, 36, 46, 56, 68, 81, 96, 116, 130, 150, 168, 190, 204, 236

Offset: 1

Hugo Pfoertner, Feb 08 2007

a(n)<= min(A127402(n),A127403(n),A127404(n)).

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

Cf. A127402, A127403, A127404, A127406. The corresponding sequences for the square lattice and hexagonal lattice are A123689 and A125851, respectively.

A127402 Number of points in a honeycomb net covered by a circular disk of diameter n if the center of the circle is chosen at the deep hole.

Original entry on oeis.org

0, 6, 6, 12, 12, 24, 24, 42, 54, 60, 72, 84, 96, 126, 138, 156, 168, 204, 204, 246, 270, 288, 312, 348, 372, 414, 450, 480, 504, 552, 564, 618, 666, 696, 744, 780, 816, 870, 930, 960, 1008, 1080, 1104, 1182, 1218, 1272, 1320, 1392, 1440, 1506, 1578, 1632

Offset: 1

Hugo Pfoertner, Feb 08 2007

nonn

			a(2)=6 because a disk of diameter 2 covers the 6 net points surrounding the deep hole.

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A127403, A127404, A127405, A127406. The corresponding sequences for the square lattice and hexagonal lattice are A053415 and A053479, respectively.

a[n_] := Sum[Boole[4*(i^2 + i*j + j^2) <= n^2 && Mod[i - j, 3] != 0], {i, -n, n}, {j, -n, n}];
Array[a, 52] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

a(n) = sum(i=-n, n, sum(j=-n, n, 4*(i^2 + i*j + j^2) <= n^2 && (i-j) % 3 != 0)); \\ Andrew Howroyd, Sep 16 2017

a(n) = 2*(A053416(n) - A127403(n)). - Andrew Howroyd, Sep 16 2017

Terms a(23) and beyond from Andrew Howroyd, Sep 16 2017

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.

cp's OEIS Frontend

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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A127405 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 minimize the number of net points covered by the disk.

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A127402 Number of points in a honeycomb net covered by a circular disk of diameter n if the center of the circle is chosen at the deep hole.

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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.

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