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A291259 Minimum number of points of the square lattice falling strictly inside a circle of radius n. 6
0, 1, 9, 25, 45, 69, 108, 145, 193, 248, 305, 373, 437, 517, 608, 697, 793, 889, 1005, 1124, 1245, 1369, 1510, 1649, 1789, 1941, 2109, 2278, 2449, 2617, 2809, 2997, 3202, 3405, 3613, 3834, 4049, 4281, 4509, 4762, 5013, 5249, 5521, 5785, 6068, 6348, 6621, 6917 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Due to the symmetry and periodicity of the square lattice it is sufficient to explore possible circles with center belonging to the triangle with vertices (0,0), (1/2,0), and (1/2,1/2).

The different regions for the centers producing constant numbers of lattice points inside circles of radius n seem to become very complex and irregular as n increases (see density plots in Links).

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 0..125

Andres Cicuttin, Plots of regions for centers of circles of several radii n enclosing constant number of lattice points

Eric Weisstein's World of Mathematics, Circle Lattice Points

FORMULA

a(n) ~ Pi*n^2.

a(n) <= A051132(n). - Joerg Arndt, Oct 03 2017

EXAMPLE

From Arkadiusz Wesolowski, Dec 18 2017 [Corrected by Andrey Zabolotskiy, Feb 19 2018]: (Start)

For a circle centered at the point (x, y) = (1/2, 0) with radius 6, there are 108 lattice points inside the circle.

Possible (but not unique) choices for the centers of the circles for radii up to 20 are given below.

.

.  Poss. center          Points in

.    x      y    Radius  the circle

.  -----  -----  ------  ----------

.    0      0       1         1

.    0      0       2         9

.    0      0       3        25

.    0      0       4        45

.    0      0       5        69

.   1/2     0       6       108

.    0      0       7       145

.    0      0       8       193

.   1/5     0       9       248

.    0      0      10       305

.    0      0      11       373

.    0      0      12       437

.    0      0      13       517

.   1/4     0      14       608

.    0      0      15       697

.    0      0      16       793

.    0      0      17       889

.    0      0      18      1005

.   1/2    1/2     19      1124

.    0      0      20      1245

(End)

MATHEMATICA

(* A291259: Minimum number of points of the square lattice falling strictly inside a circle of radius n. *)

(* The three vertices of the Explorative Triangle (ET) *)

P1={0, 0}; P2={1/2, 0}; P3={1/2, 1/2};

dd2=SquaredEuclideanDistance;

(* candidatePointQ[p, n] gives True if "p" is a candidate point, and False otherwise. A candidate point is a point belonging to a circle of radius "n" with center in the ET *)

candidatePointQ[p_, n_] := With[{dds={dd2[p, P1], dd2[p, P2], dd2[p, P3]}}, Max[dds]>=n^2>=Min[dds]];

(* Check if point "p" falls inside any circle with radius "n" and center in the ET *)

innerPointQ[p_, n_] := With[{dds={dd2[p, P1], dd2[p, P2], dd2[p, P3]}}, Max[dds]<n^2];

(* The function "candidatePoints[n]" gives the list of points with distance "n" from some point of the ET *)

candidatePoints[n_] := Select[Table[{i, j}, {i, -n, n+1}, {j, -n, n+1}]//Flatten[#, 1]&, candidatePointQ[#, n]&];

(* The function "centersFromTwoPoints[{{x1, y1}, {x2, y2}}, n]" gives the centers of the two circles with radius "n" and tangent to the pair of points {x1, y1} and {x2, y2} *) (* Note: if the distance between the two points is less than 2n then the coordinates of the centers are not integers *)

centersFromTwoPoints[{{x1_, y1_}, {x2_, y2_}}, n_] :=

Which[x1==x2,

Block[{sqrtTerm=Sqrt[4*n^2-(y1-y2)^2]/2}, {{x1-sqrtTerm, (y1+y2)/2},

{x1+sqrtTerm, (y1+y2)/2}}],

y1==y2,

Block[{sqrtTerm=Sqrt[4*n^2-(x1-x2)^2]/2}, {{(x1+x2)/2, -sqrtTerm+y1},

{(x1+x2)/2, sqrtTerm+y1}}], True,

Block[{ddxy2=dd2[{x1, y1}, {x2, y2}], sqrtTerm}, sqrtTerm=Sqrt[-(ddxy2*(-4*n^2+ddxy2)*(y1-y2)^2)]; Table[{((x1+x2)*ddxy2-sqrtTerm)/(2*ddxy2), (ddxy2*(y1^2-y2^2)+sign*(x1-x2)*sqrtTerm)/(2*ddxy2*(y1-y2))}, {sign, {1, -1}}]]];

(* The function "explorativeCenters[pairc, n]" selects the centers of circles of radius "n" of the list "pairc" laying inside the ET *)

explorativeCenters[pairc_, n_] := Select[Table[centersFromTwoPoints[pair, n], {pair, pairc}]//Flatten[#, 1]&, 0<=#[[1]]<=1/2 && 0<= #[[2]]<=#[[1]]&];

a[n_] := If[n==0, 0, Module[{points, pairc, expcent, innerpoints},

points = candidatePoints[n];

pairc = Select[Subsets[points, {2}], dd2@@#<4n^2&];

expcent = explorativeCenters[pairc, n];

innerpoints = Count[Table[{i, j}, {i, -n, n+1}, {j, -n, n+1}]//Flatten[#, 1]&, _?(innerPointQ[#, n]&)];

Min[Table[Count[points, _?(dd2[#, center]<n^2&)], {center, expcent}]] + innerpoints]];

Table[a[n], {n, 0, 20}] (* Andres Cicuttin & Andrey Zabolotskiy, Nov 14 2017 *)

CROSSREFS

Cf. A046109, A051132, A295344.

Sequence in context: A114057 A227518 A031036 * A051132 A247687 A075026

Adjacent sequences:  A291256 A291257 A291258 * A291260 A291261 A291262

KEYWORD

nonn

AUTHOR

Andres Cicuttin, Aug 21 2017

EXTENSIONS

More terms from Andrey Zabolotskiy, Nov 17 2017

STATUS

approved

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Last modified April 24 16:26 EDT 2019. Contains 322430 sequences. (Running on oeis4.)