A355479 -id:A355479 - 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.

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

Paolo Xausa, Jul 18 2022

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?

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

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

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

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 *)

A357434 a(n) is the number of distinct Q-toothpicks after the n-th stage of the structure described in A211000.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 17, 18, 18, 18, 18, 19, 20, 21, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 23, 24, 25, 26, 27, 28, 28

Offset: 0

Paolo Xausa, Sep 28 2022

nonn

See A211000 for additional information.

For the definition of Q-toothpicks, see A187210.

			In the following diagrams the A211000 structure is shown at the end of the n-th stage (Q-toothpicks are depicted as straight lines instead of circle arcs).
.
n       0       1      10      15      32      39      60      65
a(n)    0       1      10      15      16      20      23      28
.
                                                                /\
                                                                \/
                                                                 \
                                                         /       /
                                                /       /\      /\
                                                \       \/      \/
              /       /\      /\      /\      /\/\    /\/\    /\/\
                        \       \       \/      \/      \/      \/
                         \      /\      /\      /\      /\      /\
                         /      \/      \/      \/      \/      \/
                        /       /\      /\      /\      /\      /\
                        \       \/      \/      \/      \/      \/
                         \      /\      /\      /\      /\      /\
                        \/      \/      \/      \/      \/      \/
.

Cf. A187210, A211000, A355479.

A357434[nmax_]:=Module[{a={0},tp={},ep1={0,0},ep2,angle=0,turn=Pi/2},Do[If[!PrimeQ[n],If[n>5&&PrimeQ[n-1],turn*=-1];angle-=turn];tp=Union[tp,{{ep1,ep2=AngleVector[ep1,angle]}}];ep1=ep2;AppendTo[a,Length[tp]],{n,0,nmax-1}];a];
A357434[100]

A357434(nmax) = my(a=List([0,1]), newtp=[[0, 0], [1, 1]], tp=Set([newtp]), turn=1, p1, p2); if(nmax==0, return([0]));for(n=1, nmax-1, p1=newtp[1]; p2=newtp[2]; if(isprime(n), newtp=[p2, [2*p2[1]-p1[1], 2*p2[2]-p1[2]]], if(n>5 && isprime(n-1), turn*=-1); newtp=[p2, [p2[1]-turn*(p1[2]-p2[2]), p2[2]+turn*(p1[1]-p2[1])]]); tp=setunion(tp, [newtp]); listput(a,length(tp))); Vec(a);
A357434(100)

from sympy import isprime
def A357434(nmax):
    newtp, a, turn = ((0, 0), (1, 1)), [0, 1], 1
    tp = {newtp}
    for n in range(1, nmax):
        p1, p2 = newtp[0], newtp[1]
        if isprime(n): # Continue straight
            newtp = (p2, (2*p2[0]-p1[0], 2*p2[1]-p1[1]))
        else:          # Turn
            if n>5 and isprime(n-1): turn *= -1
            newtp = (p2, (p2[0]-turn*(p1[1]-p2[1]), p2[1]+turn*(p1[0]-p2[0])))
        tp.add(newtp)
        a.append(len(tp))
    return a[:nmax+1]
print(A357434(100))

A355480 a(n) is the number of distinct, hexagonal-tiled regions after the n-th step of the walk described in A355478.

Original entry on oeis.org

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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Paolo Xausa, Jul 21 2022

See A355478 for additional information and animations.

			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                2
                                         __
                                      __/ 2\*_
      *      __*   __    __          / 2\__/  \__
                     \     \__       \__/ 2\__   \__
                     /     /  \__       \__/ 2\__/  \__
                     \     \*_   \__       \__/  \__   \__
                     /     / 1\     \            / 1\     \
                     \     \__/   __/            \__/   __/
                     /     /   __/               /   __/
                     \*    \__/                  \__/
.

Paolo Xausa, Table of n, a(n) for n = 0..9999
Paolo Xausa, Illustration of selected terms up to n = 11000
Index entries for sequences related to walks

Cf. A174313, A211020, A233399, A355478, A355479.

A355480[nterms_]:=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,Length[ConnectedComponents[Graph[Flatten[cells]]]]],{n,nterms-1}];Take[a,nterms]];
A355480[100]

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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.

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A357434 a(n) is the number of distinct Q-toothpicks after the n-th stage of the structure described in A211000.

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A355480 a(n) is the number of distinct, hexagonal-tiled regions after the n-th step of the walk described in A355478.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica