A355478 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.

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

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 n-th step.

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

Links

Paolo Xausa, Table of n, a(n) for n = 0..9999

Paolo Xausa, Animation of terms n = 0..40

Paolo Xausa, Animation of terms n = 0..749

Paolo Xausa, Illustration of selected terms up to n = 11000

Index entries for sequences related to walks

Example

In the following diagrams the walk is shown at the end of the n-th 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];
A355478[100] (* Paolo Xausa, Jan 04 2023 *)

Crossrefs

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

Sequence in context: A046042 A287866 A071841 * A097876 A177830 A111859

Adjacent sequences: A355475 A355476 A355477 * A355479 A355480 A355481

Keyword

nonn,walk

Author

Paolo Xausa, Jul 18 2022

Status

approved