cp's OEIS Frontend

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.

%I A355478 #35 Jan 05 2023 10:19:18
%S A355478 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 A355478 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 A355478 6,6,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9
%N 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.
%C A355478 At step 0, the honeybee is at the origin. No honeycomb cell wall is yet built.
%C A355478 At step 1, the honeybee walks one unit eastward, building the first cell wall.
%C A355478 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 A355478 a(n) is the number of distinct, "unit" honeycomb cells (six sides of unit length) built after the n-th step.
%C A355478 Does this walk generate a full hexagonal tiling of the plane?
%H A355478 Paolo Xausa, <a href="/A355478/b355478.txt">Table of n, a(n) for n = 0..9999</a>
%H A355478 Paolo Xausa, <a href="/A355478/a355478_2.gif">Animation of terms n = 0..40</a>
%H A355478 Paolo Xausa, <a href="/A355478/a355478_3.gif">Animation of terms n = 0..749</a>
%H A355478 Paolo Xausa, <a href="/A355478/a355478_1.pdf">Illustration of selected terms up to n = 11000</a>
%H A355478 <a href="/index/Wa#WALKS">Index entries for sequences related to walks</a>
%e A355478 In the following diagrams the walk is shown at the end of the n-th step, together with the position of the bee (*).
%e A355478 .
%e A355478 n     0      1      8        28               60
%e A355478 a(n)  0      0      0         1                5
%e A355478                                          __
%e A355478                                       __/ 5\*_
%e A355478       *      __*   __    __          / 4\__/  \__
%e A355478                      \     \__       \__/ 3\__   \__
%e A355478                      /     /  \__       \__/ 2\__/  \__
%e A355478                      \     \*_   \__       \__/  \__   \__
%e A355478                      /     / 1\     \            / 1\     \
%e A355478                      \     \__/   __/            \__/   __/
%e A355478                      /     /   __/               /   __/
%e A355478                      \*    \__/                  \__/
%e A355478 .
%t A355478 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 A355478[100] (* _Paolo Xausa_, Jan 04 2023 *)
%Y A355478 Cf. A174313, A211020, A233399, A355479, A355480, A359529.
%K A355478 nonn,walk
%O A355478 0,37
%A A355478 _Paolo Xausa_, Jul 18 2022