A053479 -id:A053479 - cp's OEIS frontend

A160826 Improvement of A125852 over A053416, A053479 and A053417.

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 4, 3, 0, 0, 0, 1, 0, 0, 0, 0, 2, 4, 5, 1, 3, 1, 0, 3, 2, 3, 4, 3, 4, 5, 6, 9, 4, 3, 0, 1, 0, 0, 0, 2, 4, 3, 4, 5, 10, 14, 3, 6, 0, 7, 0, 4, 5, 1, 8, 6, 0, 4, 7, 8, 6, 5, 11, 5, 9, 12, 12, 4, 0, 11, 7, 12, 0, 3, 1, 0, 1, 5, 0, 6, 2, 10, 11, 25, 17, 3, 2, 0, 9, 0, 12, 5, 0, 4, 2

Offset: 1

Hagen von Eitzen, May 27 2009

nonn

How many more lattice points of a hexagonal lattice can be covered by placing a disk of diameter n at an optimal center instead of one of the three obvious centers (a lattice point, midpoint between two lattice points, barycenter of a fundamental triangle)?
The first difference occurs at n=9, when a diameter 9 disc around e.g. (1/2, 4*sqrt(5)) covers more lattice points than one around (0,0) or (1/2,0) or (1/2,sqrt(3)/6).
Clearly a(n) = O(n) as all "extra" points have norm approximately n^2/4 if the optimal center is chosen near (0,0). Does a(n)/n converge? Are there only finitely many n with a(n)=0?

			For diameters n=2,4,6,8 a disc around (0,0) and for n=1,3,5,7 a disc around(1/2,0) happens to be optimal (covers as many points as possible); therefore a(1)=a(2)=...=a(8)=0.
a(9) = A125852(9) - max(A053416(9),A053479(9),A053417(9)) = 77 - max(73,69,76) = 1.

H. v. Eitzen, Table of n, a(n) for n=1..1000

a(n) = A125852(n) - max(A053416(n),A053479(n),A053417(n))

A053417 Circle numbers (version 5): a(n) = number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (1/2,0).

Original entry on oeis.org

0, 2, 4, 10, 14, 24, 30, 48, 60, 76, 92, 110, 130, 154, 178, 208, 230, 264, 288, 330, 364, 406, 442, 482, 522, 564, 614, 664, 712, 766, 812, 874, 922, 990, 1050, 1112, 1176, 1240, 1312, 1382, 1452, 1530, 1598, 1684, 1750, 1840, 1920, 2008, 2092, 2182, 2266

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 10 2000

Equivalently, number of points in a hexagonal lattice covered by a circular disk of diameter n if the center of the circle is chosen at the middle between two lattice points. - Hugo Pfoertner, Jan 07 2007

Same as above but "number of disks (r = 1)" instead of "number of points". a(2^n - 1) = A239073(n), n >= 1. See illustration in links. - Kival Ngaokrajang, Apr 06 2014

H. v. Eitzen, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A053411, A053414, A053415, A053416, A053417.
Cf. A053479, A125851, A125852.
Cf. A239073.

a[n_] := Sum[dj = Sqrt[Abs[4*n^2 + 6*i - 3*i^2 - 3]]/4; j1 = (1 - 2*i)/4 - dj // Floor; j2 = (1 - 2*i)/4 + dj // Ceiling; Sum[ Boole[i^2 - i - j/2 + i*j + j^2 + 1/4 <= n^2/4], {j, j1, j2}], {i, -n - 1, n + 3}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 06 2013 *)

a(n)/(n/2)^2 -> Pi*2/sqrt(3).

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

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.

Original entry on oeis.org

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

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.

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

cp's OEIS Frontend

A160826 Improvement of A125852 over A053416, A053479 and A053417.

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A053417 Circle numbers (version 5): a(n) = number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (1/2,0).

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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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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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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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