

A123689


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 minimum possible number of lattice points.


5



0, 2, 4, 10, 16, 26, 32, 46, 60, 74, 88, 108, 124, 146, 172, 194, 216, 248, 276, 308
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OFFSET

1,2


COMMENTS

a(n)<=min(A053411(n),A053414(n),A053415(n)).
Using brute force computation and a step size of 1/1000 (though 1/200 suffices), the [conjectured] terms a(21) to a(40) would be: 332, 374, 408, 438, 484, 522, 560, 608, 648, 698, 740, 794, 846, 894, 952, 1006, 1060, 1124, 1184, 1248.  JeanFrançois Alcover, Jan 08 2018


LINKS

Table of n, a(n) for n=1..20.
Hugo Pfoertner, Minimal number of points in the square lattice covered by circular disks. Illustrations.


EXAMPLE

a(1)=0: Circle with diameter 1 with center (0.5,0.5) covers no lattice points; a(2)=2: Circle with diameter 2 with center (0,eps) covers 2 lattice points;
a(3)=4: Circle with diameter 3 with center (0.5,0.5) covers 4 lattice points.


MATHEMATICA

dx = 1/200; y0 = 0; (* To speed up computation, the step size dx is experimentally adjusted and the circle center is taken on the xaxis. *)
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, {x, y0}, r], {x, y0}}, {x, 1/2, 1/2, dx}] // Sort // First; Print["a(", n, ") = ", an, ", center = ", center // InputForm]; an];
Table[a[n], {n, 1, 20}] (* JeanFrançois Alcover, Jan 08 2018 *)


CROSSREFS

Cf. A123690, A053411, A053414, A053415, A122224, A291259.
The corresponding sequences for the hexagonal lattice and the honeycomb net are A125851 and A127405, respectively.
Sequence in context: A218665 A189558 A111149 * A137928 A293154 A144834
Adjacent sequences: A123686 A123687 A123688 * A123690 A123691 A123692


KEYWORD

more,nonn


AUTHOR

Hugo Pfoertner, Oct 09 2006


STATUS

approved



