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

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

Offset: 0

Author

Andres Cicuttin, Aug 21 2017

Keywords

• nonn

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).

Examples

			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)
		

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

Crossrefs

Cf. A046109, A051132, A295344.

Programs

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]Andres Cicuttin & Andrey Zabolotskiy, Nov 14 2017 *)

Formula

a(n) ~ Pi*n^2.

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

Extensions

More terms from Andrey Zabolotskiy, Nov 17 2017