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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. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..51.

Hugo Pfoertner, Maximum number of points in the square lattice covered by circular disks. Illustrations.

EXAMPLE

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.

MATHEMATICA

dx[n_] := If[n < 20, 1/24, 1/400]; (* step dx adjusted to speed-up computation *)

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, {x0, y0}, r], {x0, y0}}, {x0, 0, 1/2, dx[n]}, {y0, 0, 1/2, dx[n]}] // Flatten[#, 1] & // Sort // Last; Print["a(", n, ") = ", an, " center = ", center // InputForm]; an];

Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Jan 02 2018, updated Jan 06 2018 *)

(* 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 *)

CROSSREFS

Cf. A123689, A053411, A053414, A053415, A122224, A295344, A291259.

The corresponding sequences for the hexagonal lattice and the honeycomb net are A125852 and A127406, respectively.

Sequence in context: A011905 A306674 A098065 * A199935 A090937 A325717

Adjacent sequences:  A123687 A123688 A123689 * A123691 A123692 A123693

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Oct 09 2006, Feb 11 2007

EXTENSIONS

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

STATUS

approved

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Last modified June 26 10:12 EDT 2019. Contains 324375 sequences. (Running on oeis4.)