A125852 -id:A125852 - 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))

A053416 Circle numbers (version 4): 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 (0,0).

Original entry on oeis.org

1, 1, 7, 7, 19, 19, 37, 43, 61, 73, 91, 109, 127, 151, 187, 199, 241, 253, 301, 313, 367, 397, 439, 475, 517, 571, 613, 661, 721, 757, 823, 859, 931, 979, 1045, 1111, 1165, 1237, 1303, 1381, 1459, 1519, 1615, 1663, 1765, 1813, 1921, 1993, 2083, 2173, 2263

Offset: 0

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

In other words, number of points in a hexagonal lattice covered by a circle of diameter n if the center of the circle is chosen at a grid point. - Hugo Pfoertner, Jan 07 2007

Same as above but "number of disks (r = 1)" instead of "number of points". 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. A003215, A125849, A125850, A125851, A125852.

Cf. A053411, A053414, A053415, A053417, A053458 (open disk), A308685 (bisection).

A053416 := proc(d)
    local a,j,imin,imax ;
    a := 0 ;
    for j from -floor(d/sqrt(3)) do
        if j^2*3 > d^2 and j> 0 then
            break ;
        end if;
        imin := ceil((-j-sqrt(d^2-3*j^2))/2) ;
        imax := floor((-j+sqrt(d^2-3*j^2))/2) ;
        a := a+imax-imin+1 ;
    end do:
    a ;
end proc:
seq(A053416(d),d=0..30) ; # R. J. Mathar, Nov 22 2022

a[n_] := Sum[Boole[4*(i^2 + i*j + j^2) <= n^2], {i, -n, n}, {j, -n, n}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 06 2013, updated Apr 08 2022 to correct a discrepancy wrt b-file noticed by Georg Fischer *)

a(n)/(n/2)^2->Pi*2/sqrt(3).
a(n) >= A053458(n). - R. J. Mathar, Nov 22 2022
a(2*n) = A308685(n). - R. J. Mathar, Nov 22 2022

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

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

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

A122226 Length of the longest possible self-avoiding path on the 2-dimensional triangular lattice such that the path fits into a circle of diameter n.

Original entry on oeis.org

1, 7, 10, 19, 24, 37, 48, 61

Offset: 1

Hugo Pfoertner, Sep 25 2006

The path may be open or closed. For larger n several solutions with the same number of segments exist.

It is conjectured that the sequence is identical with A125852 for all n>1. That means that it is always possible to find an Hamiltonian cycle on the maximum possible number of lattice points that can be covered by circular disks of diameter >=2. For the given additional terms it was easily possible to construct such closed paths by hand, using the lattice subset found by the exhaustive search for A125852. See the examples at the end of the linked pdf file a122226.pdf that were all generated without using a program. - Hugo Pfoertner, Jan 12 2007

Hugo Pfoertner, Examples of compact self avoiding paths on a triangular lattice.

Cf. A003215, A004016; A125852 gives upper bounds for a(n).

Cf. A122223, A122224.

a(7) and a(8) from Hugo Pfoertner, Dec 11 2006

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.

Original entry on oeis.org

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.

A053479 Circle numbers (version 6): a(n) = number of points (i+j/2,jsqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (1/2, 1/(2sqrt(3))).

Original entry on oeis.org

0, 0, 3, 6, 12, 21, 30, 42, 54, 69, 90, 102, 129, 150, 174, 198, 225, 258, 288, 327, 354, 396, 435, 471, 522, 558, 609, 654, 702, 759, 807, 864, 924, 981, 1038, 1104, 1173, 1230, 1308, 1368, 1443, 1512, 1590, 1671, 1746, 1830, 1908, 2001, 2076, 2166, 2265

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 14 2000

In other words, 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 deep hole. - Hugo Pfoertner, Jan 07 2007

Also number of integer coordinate pairs (s,t) satisfying s^2+t^2+st-s-t <= n^2/4-1/3. The a(2)=3 coordinate pairs are (s,t)=(0,0), (0,1) and (1,0). The a(3)=6 coordinate pairs are (-1,1),(0,0),(0,1),(1,-1),(1,0) and (1,1). - R. J. Mathar, Feb 23 2007

Cf. A053411, A053414, A053415, A053416, A053417.

Cf. A005882, A125850, A125851, A125852.

A053479 := proc(n) local res,a,b ; res :=0 ; for a from -n to n do for b from -n to n do if a^2+b^2+a*b-a-b <= n^2/4-1/3 then res := res+1 ; fi ; od ; od ; RETURN(res) ; end : for n from 1 to 40 do printf("%d ",A053479(n)) ; od ; # R. J. Mathar, Feb 23 2007

cx = 1/2; cy = 1/(2*Sqrt[3]); a[n_] := Sum[ dj = (1/2)* Sqrt[Abs[-3*cx^2 + 2*Sqrt[3]*cx*cy - cy^2 + 6*cx*i - 2*Sqrt[3]*cy*i - 3*i^2 + n^2]]; j1 = cx/2 + (Sqrt[3]*cy)/2 - i/2 - dj // Floor ; j2 = cx/2 + (Sqrt[3]*cy)/2 - i/2 + dj // Ceiling; Sum[Boole[(i + j/2 - cx)^2 + (j*(Sqrt[3]/2) - cy)^2 <= n^2/4], {j, j1, j2}], {i, -(n + 1)/2 - 2 // Floor, (n + 1)/2 + 3 // Ceiling}]; 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

A346126 Numbers m such that no self-avoiding walk of length m + 1 on the hexagonal lattice fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 10, 12, 14, 15, 16, 19, 20, 22, 23, 24, 25, 27, 31, 32, 34, 37, 38, 39, 40, 42, 43, 44, 45, 48, 49, 55, 56, 57, 58, 60, 61

Offset: 1

Hugo Pfoertner and Markus Sigg, Jul 31 2021

Open and closed walks are allowed. It is conjectured that all optimal paths are closed except for the trivial path of length 1. See the related conjecture in A122226.

			See link for illustrations of terms corresponding to diameters D <= 8.

Hugo Pfoertner, Examples of paths of maximum length.

Cf. A122226, A125852, A127399, A127400, A127401, A151541, A284869, A306176, A316196.
Cf. A346123 (similar to this sequence, but for honeycomb net), A346124 (ditto for square lattice).
Cf. A346125, A346127-A346132 (similar to this sequence, but with other sets of turning angles).

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

A160826 Improvement of A125852 over A053416, A053479 and A053417.

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A053416 Circle numbers (version 4): 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 (0,0).

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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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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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A122226 Length of the longest possible self-avoiding path on the 2-dimensional triangular lattice such that the path fits into a circle of diameter n.

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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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A053479 Circle numbers (version 6): 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, 1/(2*sqrt(3))).

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A346126 Numbers m such that no self-avoiding walk of length m + 1 on the hexagonal lattice fits into the smallest circle that can enclose a walk of length m.

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A053479 Circle numbers (version 6): a(n) = number of points (i+j/2,jsqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (1/2, 1/(2sqrt(3))).