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

Author

Paolo Xausa, Jul 18 2022

Keywords

• nonn
• walk

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?

Examples

			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\     \
                     \     \__/   __/            \__/   __/
                     /     /   __/               /   __/
                     \*    \__/                  \__/
.
		

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

Crossrefs

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

Programs

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 *)