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Numerators of minimal squared radii of circular disks covering a record number of lattice points of the hexagonal lattice, when the centers of the circles are chosen to maximize the number of covered lattice points.
3

%I #10 Aug 09 2021 04:38:07

%S 0,1,1,3,1,7,49,9,7,13,169,91,4,133,21,361,1729,169,19,7,961,133,9,39,

%T 21793,481,31,9331,301,3367,49,817,13,361,931,1813,63,16

%N Numerators of minimal squared radii of circular disks covering a record number of lattice points of the hexagonal lattice, when the centers of the circles are chosen to maximize the number of covered lattice points.

%C It is conjectured that the number of covered grid points is given by A346126(n-1) for n>2.

%H Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a346126.htm">Examples of self avoiding walks of minimum diameter on the hexagonal lattice</a>.

%e 0, 1/4, 1/3, 3/4, 1, 7/4, 49/25, 9/4, 7/3, 13/4, 169/48, 91/25, 4, 133/27, 21/4, 361/64, 1729/289, 169/27, 19/3, 7, 961/121, 133/16, 9, 39/4, 21793/2187, ...

%e .

%e Diameter Covered R^2 =

%e of disk grid (D/2)^2 =

%e n D points a(n) / A346785(n)

%e .

%e 1 0.00000 1 0 / 1

%e 2 1.00000 2 1 / 4

%e 3 1.15470 3 1 / 3

%e 4 1.73205 4 3 / 4

%e 5 2.00000 7 1 / 1

%e 6 2.64575 8 7 / 4

%e 7 2.80000 9 49 / 25

%e 8 3.00000 10 9 / 4

%e 9 3.05505 12 7 / 3

%e 10 3.60555 14 13 / 4

%Y Corresponding denominators are A346785.

%Y Cf. A125852, A346126.

%K nonn,frac,more

%O 1,4

%A _Hugo Pfoertner_, Aug 08 2021