A123689 - OEIS

%I #19 Mar 02 2018 09:28:32

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

%N 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.

%C a(n)<=min(A053411(n),A053414(n),A053415(n)).

%C 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. - _Jean-François Alcover_, Jan 08 2018

%H Hugo Pfoertner, <a href="/A123689/a123689.pdf">Minimal number of points in the square lattice covered by circular disks.</a> Illustrations.

%e 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;

%e a(3)=4: Circle with diameter 3 with center (0.5,0.5) covers 4 lattice points.

%t dx = 1/200; y0 = 0; (* To speed up computation, the step size dx is experimentally adjusted and the circle center is taken on the x-axis. *)

%t cnt[pts_, ctr_, r_] := Count[pts, pt_ /; Norm[pt - ctr] <= r];

%t 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];

%t Table[a[n], {n, 1, 20}] (* _Jean-François Alcover_, Jan 08 2018 *)

%Y Cf. A123690, A053411, A053414, A053415, A122224, A291259.

%Y The corresponding sequences for the hexagonal lattice and the honeycomb net are A125851 and A127405, respectively.

%K more,nonn

%O 1,2

%A _Hugo Pfoertner_, Oct 09 2006