

A355478


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


6



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, 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, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9
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OFFSET

0,37


COMMENTS

At step 0, the honeybee is at the origin. No honeycomb cell wall is yet built.
At step 1, the honeybee walks one unit eastward, building the first cell wall.
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).
a(n) is the number of distinct, "unit" honeycomb cells (six sides of unit length) built after the nth step.
Does this walk generate a full hexagonal tiling of the plane?


LINKS



EXAMPLE

In the following diagrams the walk is shown at the end of the nth step, together with the position of the bee (*).
.
n 0 1 8 28 60
a(n) 0 0 0 1 5
__
__/ 5\*_
* __* __ __ / 4\__/ \__
\ \__ \__/ 3\__ \__
/ / \__ \__/ 2\__/ \__
\ \*_ \__ \__/ \__ \__
/ / 1\ \ / 1\ \
\ \__/ __/ \__/ __/
/ / __/ / __/
\* \__/ \__/
.


MATHEMATICA

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];


CROSSREFS



KEYWORD

nonn,walk


AUTHOR



STATUS

approved



