A123690 - OEIS

%I #43 Jun 11 2022 04:24:44

%S 2,5,9,14,22,32,41,52,69,81,97,116,137,157,180,208,231,258,293,319,

%T 351,384,421,457,495,540,578,623,667,716,761,812,861,914,973,1025,

%U 1085,1142,1201,1268,1328,1396,1460,1528,1597,1669,1745,1816,1893,1976,2053

%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 maximum possible number of lattice points.

%C a(n) >= max(A053411(n), A053414(n), A053415(n)).

%C 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).

%H Hugo Pfoertner, <a href="/A123690/a123690.pdf">Maximum number of points in the square lattice covered by circular disks.</a> Illustrations.

%e a(1)=2: Circle with diameter 1 and center (0,0.5) covers 2 lattice points;

%e a(2)=5: Circle with diameter 2 and center (0,0) covers 5 lattice points;

%e a(3)=4: Circle with diameter 3 and center (0,0) covers 9 lattice points;

%e a(4)=14: Circle with diameter 4 and center (0.5,0.2) covers 14 lattice points.

%t (* An exact program using the functions from A291259: *)

%t Clear[a]; a[n_] := Module[{points, pairc, expcent, innerpoints, cn=Ceiling[n], allpairs},

%t allpairs = Flatten[Table[{i, j}, {i, -cn, cn+1}, {j, -cn, cn+1}], 1];

%t points = Select[allpairs, candidatePointQ[#, n]&];

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

%t expcent = explorativeCenters[pairc, n];

%t innerpoints = Count[allpairs, _?(innerPointQ[#, n]&)];

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

%t Table[a[n/2], {n, 20}] (* _Andrey Zabolotskiy_, Feb 21 2018 *)

%Y Cf. A123689, A053411, A053414, A053415, A122224, A295344, A291259, A346993, A346994, A346995.

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

%K nonn

%O 1,1

%A _Hugo Pfoertner_, Oct 09 2006, Feb 11 2007

%E 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