A355478 - OEIS

%I #35 Jan 05 2023 10:19:18

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,

%T 1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,

%U 6,6,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9

%N The honeybee prime walk: a(n) is the number of closed honeycomb cells after the n-th step of the walk described in the comments.

%C At step 0, the honeybee is at the origin. No honeycomb cell wall is yet built.

%C At step 1, the honeybee walks one unit eastward, building the first cell wall.

%C At step n, the honeybee turns 60 degrees clockwise or counterclockwise (depending on whether n is prime or not, respectively), then walks one unit in the new direction, building the next cell wall (which may coincide with an existing wall).

%C a(n) is the number of distinct, "unit" honeycomb cells (six sides of unit length) built after the n-th step.

%C Does this walk generate a full hexagonal tiling of the plane?

%H Paolo Xausa, <a href="/A355478/b355478.txt">Table of n, a(n) for n = 0..9999</a>

%H Paolo Xausa, <a href="/A355478/a355478_2.gif">Animation of terms n = 0..40</a>

%H Paolo Xausa, <a href="/A355478/a355478_3.gif">Animation of terms n = 0..749</a>

%H Paolo Xausa, <a href="/A355478/a355478_1.pdf">Illustration of selected terms up to n = 11000</a>

%H <a href="/index/Wa#WALKS">Index entries for sequences related to walks</a>

%e In the following diagrams the walk is shown at the end of the n-th step, together with the position of the bee (*).

%e .

%e n     0      1      8        28               60

%e a(n)  0      0      0         1                5

%e                                          __

%e                                       __/ 5\*_

%e       *      __*   __    __          / 4\__/  \__

%e                      \     \__       \__/ 3\__   \__

%e                      /     /  \__       \__/ 2\__/  \__

%e                      \     \*_   \__       \__/  \__   \__

%e                      /     / 1\     \            / 1\     \

%e                      \     \__/   __/            \__/   __/

%e                      /     /   __/               /   __/

%e                      \*    \__/                  \__/

%e .

%t A355478[nmax_]:=Module[{a={0}, walk={{0, 0}}, angle=0, cells}, Do[AppendTo[walk, AngleVector[Last[walk], angle+=If[PrimeQ[n], -1, 1]Pi/3]]; cells=FindCycle[Graph[MapApply[UndirectedEdge, Partition[walk, 2, 1]]], {6}, All]; AppendTo[a, CountDistinct[Map[Sort, Map[First, cells, {2}]]]], {n, nmax}]; a];

%t A355478[100] (* _Paolo Xausa_, Jan 04 2023 *)

%Y Cf. A174313, A211020, A233399, A355479, A355480, A359529.

%K nonn,walk

%O 0,37

%A _Paolo Xausa_, Jul 18 2022