A127405 -id:A127405 - 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.

A125851 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 minimum possible number of lattice points.

Original entry on oeis.org

0, 3, 6, 12, 19, 30, 40, 54, 69, 87, 102, 123, 149, 174, 198, 225, 253, 287, 313, 354, 396, 435

Offset: 1

Hugo Pfoertner, Jan 07 2007, Feb 11 2007

a(n)<=min(A053416(n),A053479(n),A053417(n))

Index entries for sequences related to A2 = hexagonal = triangular lattice
Hugo Pfoertner, Minimal number of points in the hexagonal lattice covered by circular disks. Illustrations.

Cf. A053416, A053479, A053417, A125852. The corresponding sequences for the square lattice and the honeycomb net are A123689 and A127405, respectively.

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 *)

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

A127403 Number of points in a honeycomb net covered by a circular disk of diameter n if the center of the circle is chosen at a grid point.

Original entry on oeis.org

1, 1, 4, 4, 13, 13, 25, 31, 40, 46, 61, 73, 85, 103, 124, 130, 163, 169, 199, 211, 244, 262, 295, 319, 343, 385, 406, 436, 481, 505, 547, 577, 622, 646, 697, 739, 775, 829, 868, 916, 979, 1015, 1075, 1111, 1174, 1204, 1285, 1333, 1387, 1453, 1510, 1558, 1639

Offset: 0

Hugo Pfoertner, Feb 08 2007

nonn

			a(2)=4 because a disk of diameter 2 covers the center of the circle and the 3 net points at distance 1.

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A127402, A127404, A127405, A127406. The corresponding sequences for the square lattice and hexagonal lattice are A053411 and A053416, respectively.

a[n_] := Sum[Boole[4*(i^2 + i*j + j^2) <= n^2 && Mod[i - j , 3] != 1], {i, -n, n}, {j, -n, n}];
Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)

a(n) = sum(i=-n, n, sum(j=-n, n, 4*(i^2 + i*j + j^2) <= n^2 && (i-j) % 3 != 1)); \\ Andrew Howroyd, Sep 16 2017

a(0) and 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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A125851 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 minimum possible number of lattice points.

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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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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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A127403 Number of points in a honeycomb net covered by a circular disk of diameter n if the center of the circle is chosen at a grid point.

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Mathematica

PARI

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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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