cp's OEIS Frontend

A123690 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.

%I A123690 #43 Jun 11 2022 04:24:44
%S A123690 2,5,9,14,22,32,41,52,69,81,97,116,137,157,180,208,231,258,293,319,
%T A123690 351,384,421,457,495,540,578,623,667,716,761,812,861,914,973,1025,
%U A123690 1085,1142,1201,1268,1328,1396,1460,1528,1597,1669,1745,1816,1893,1976,2053
%N A123690 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 A123690 a(n) >= max(A053411(n), A053414(n), A053415(n)).
%C A123690 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 A123690 Hugo Pfoertner, <a href="/A123690/a123690.pdf">Maximum number of points in the square lattice covered by circular disks.</a> Illustrations.
%e A123690 a(1)=2: Circle with diameter 1 and center (0,0.5) covers 2 lattice points;
%e A123690 a(2)=5: Circle with diameter 2 and center (0,0) covers 5 lattice points;
%e A123690 a(3)=4: Circle with diameter 3 and center (0,0) covers 9 lattice points;
%e A123690 a(4)=14: Circle with diameter 4 and center (0.5,0.2) covers 14 lattice points.
%t A123690 (* An exact program using the functions from A291259: *)
%t A123690 Clear[a]; a[n_] := Module[{points, pairc, expcent, innerpoints, cn=Ceiling[n], allpairs},
%t A123690 allpairs = Flatten[Table[{i, j}, {i, -cn, cn+1}, {j, -cn, cn+1}], 1];
%t A123690 points = Select[allpairs, candidatePointQ[#, n]&];
%t A123690 pairc = Select[Subsets[points, {2}], dd2@@#<=4n^2&];
%t A123690 expcent = explorativeCenters[pairc, n];
%t A123690 innerpoints = Count[allpairs, _?(innerPointQ[#, n]&)];
%t A123690 Max[Table[Count[points, _?(dd2[#, center]<=n^2&)], {center, expcent}]] + innerpoints];
%t A123690 Table[a[n/2], {n, 20}] (* _Andrey Zabolotskiy_, Feb 21 2018 *)
%Y A123690 Cf. A123689, A053411, A053414, A053415, A122224, A295344, A291259, A346993, A346994, A346995.
%Y A123690 The corresponding sequences for the hexagonal lattice and the honeycomb net are A125852 and A127406, respectively.
%K A123690 nonn
%O A123690 1,1
%A A123690 _Hugo Pfoertner_, Oct 09 2006, Feb 11 2007
%E A123690 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