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 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. 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 (list; graph; refs; listen; history; text; internal format)
 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: A054900 A046042 A071841 * A097876 A177830 A111859 Adjacent sequences: A355475 A355476 A355477 * A355479 A355480 A355481 KEYWORD nonn,walk AUTHOR Paolo Xausa, Jul 18 2022 STATUS approved

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Last modified November 29 07:03 EST 2023. Contains 367429 sequences. (Running on oeis4.)