A127406 -id:A127406 - cp's OEIS frontend

A125852 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 maximum possible number of lattice points.

2, 7, 10, 19, 24, 37, 48, 61, 77, 94, 115, 134, 157, 187, 208, 241, 265, 301, 330, 367, 406, 444, 486, 527, 572, 617, 665, 721, 769, 825, 877, 935, 993, 1054, 1117, 1182, 1249, 1316, 1385, 1459, 1531, 1615, 1684, 1765, 1842, 1925, 2011, 2096, 2187, 2276

Offset: 1

Hugo Pfoertner, Jan 07 2007, Feb 11 2007

nonn

a(n)>=max(A053416(n),A053479(n),A053417(n)). a(n) is an upper bound for the number of segments of a self avoiding path on the 2-dimensional triangular lattice such that the path fits into a circle of diameter n. A122226(n)<=a(n).

H. v. Eitzen, Table of n, a(n) for n=1..1000
Index entries for sequences related to A2 = hexagonal = triangular lattice
Hugo Pfoertner, Maximum number of points in the hexagonal lattice covered by circular disks. Illustrations.

Cf. A053416, A053479, A053417, A125851, A122226. The corresponding sequences for the square lattice and the honeycomb net are A123690 and A127406, respectively.

More terms copied from b-file by Hagen von Eitzen, Jun 17 2009

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

A122223 Length of the longest possible self avoiding path on the 2-dimensional honeycomb net such that the path fits into a circle of diameter n.

Original entry on oeis.org

1, 6, 6, 12, 15, 23, 33, 42, 53, 62, 74, 90, 103

Offset: 1

Hugo Pfoertner, Sep 25 2006, Feb 11 2007

The path may be open or closed. For larger n several solutions with the same number of segments exist. An upper bound for the number of segments is given by A127406(n).

Hugo Pfoertner, Examples of compact self avoiding paths on a honeycomb net.

Cf. A122224, A122226, A127406.

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

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

A125852 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 maximum possible number of lattice points.

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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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A122223 Length of the longest possible self avoiding path on the 2-dimensional honeycomb net such that the path fits into a circle of diameter n.

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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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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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