cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A355479 a(n) is the number of distinct honeycomb cell walls built after the n-th step of the walk described in A355478.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 20, 20, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 30, 31, 31, 31, 31, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 46, 46, 46, 47, 47, 48, 49, 50, 51, 51, 51
Offset: 0

Views

Author

Paolo Xausa, Jul 18 2022

Keywords

Comments

See A355478 for more information, animations, and illustration of selected terms.

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      1      8        24               47
                                         __
                                      __/  \*_
      *      __*   __    __          /  \__/  \__
                     \     \__       \__/  \__   \__
                     /     /  \__       \__/  \__/  \__
                     \     \*_   \__       \__/  \__   \__
                     /     /  \     \            /  \     \
                     \     \__/   __/            \__/   __/
                     /     /   __/               /   __/
                     \*    \__/                  \__/
.

Links

Crossrefs

Cf. A174313, A211020, A233399, A355478, A355480, A357434.

Programs

  • Mathematica
    A355479[nmax_]:=Module[{a={0},w={},p1={0, 0},p2,angle=0},Do[w=Union[w,{Sort[{p1,p2=AngleVector[p1,angle+=If[PrimeQ[n],-1,1]Pi/3]}]}];p1=p2;AppendTo[a,Length[w]],{n,nmax}];a];
A355479[100] (* Paolo Xausa, Jan 05 2023 *)

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

Views

Author

Paolo Xausa, Jul 21 2022

Keywords

Comments

See A355478 for additional information and animations.

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

Links

Crossrefs

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

Programs

  • Mathematica
    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]

A359529 Indices of records in A355478.

Original entry on oeis.org

0, 28, 36, 52, 57, 60, 65, 70, 78, 88, 93, 97, 100, 134, 138, 143, 156, 161, 164, 170, 191, 214, 255, 261, 275, 288, 297, 310, 322, 341, 358, 363, 371, 377, 384, 399, 448, 453, 472, 483, 496, 503, 508, 524, 588, 635, 656, 660, 665, 680, 713, 720, 737, 746, 755
Offset: 1

Views

Author

Paolo Xausa, Jan 04 2023, following a suggestion from Peter Munn

Keywords

Links

Crossrefs

Cf. A355478.

Programs

  • Mathematica
    A355478[nmax_]:=Module[{walk={{0,0}},angle=0,cells},Join[{0},Table[AppendTo[walk,AngleVector[Last[walk],angle+=If[PrimeQ[n],-1,1]Pi/3]];cells=FindCycle[Graph[MapApply[UndirectedEdge,Partition[walk,2,1]]],{6},All];CountDistinct[Map[Sort,Map[First,cells,{2}]]],{n,nmax}]]];
Flatten[Join[{0},Position[Differences[A355478[500]],_?Positive]]]

A211000 Coordinates (x,y) of the endpoint of a structure (or curve) formed by Q-toothpicks in which the inflection points are the prime numbers A000040.

Original entry on oeis.org

0, 0, 1, 1, 2, 0, 3, -1, 4, -2, 3, -3, 2, -4, 3, -5, 4, -6, 3, -7, 2, -6, 3, -5, 4, -4, 3, -3, 2, -2, 3, -1, 4, -2, 3, -3, 2, -4, 3, -5, 4, -6, 3, -7, 2, -6, 3, -5, 4, -4, 3, -3, 2, -4, 3, -5, 4, -4, 3, -3, 2, -2, 3, -1, 4, 0, 3, 1, 2, 0, 3, -1, 4, 0
Offset: 0

Views

Author

Omar E. Pol, Mar 30 2012

Keywords

Comments

On the infinite square grid the structure looks like a column of tangent circles of radius 1. The structure arises from the prime numbers A000040. The behavior seems to be as modular arithmetic but in a growing structure. The values on the axis "x" are easy to predict (see A211010). On the other hand the values on the axis "y" do not seem to be predictable (see A211011). This is a member of the family of the structures or curves mentioned in A210838. The odd numbers > 1 are located on the main axis of the structure. Note that here the Q-toothpicks can be superposed. For the definition of Q-toothpicks see A187210. A211021 gives the number of stage where a new circle appears in the structure. For the number of circles after the n-th stage see A211020. For the location of the centers of the circles see A211022. For the sums of the visible nodes after the n-th stage see A211024.

Examples

			We start at stage 0 with no Q-toothpicks.
At stage 1 we place a Q-toothpick centered at (1,0) with its endpoints at (0,0) and (1,1).
At stage 2 we place a Q-toothpick centered at (1,0) with its endpoints at (1,1) and (2,0). Since 2 is a prime number we have that the end of the curve is also an inflection point.
At stage 3 we place a Q-toothpick centered at (3,0) with its endpoints at (2,0) and (3,-1). Since 3 is a prime number we have that the end of the curve is also an inflection point.
At stage 4 we place a Q-toothpick centered at (3,-2) with its endpoints at (3,-1) and (4,-2).
-------------------------------------
.                    The end as
.          Pair      inflection
n        (x    y)      point
-------------------------------
0         0,   0,        -
1         1,   1,        -
2         2,   0,       Yes
3         3,  -1,       Yes
4         4,  -2,        -
5         3,  -3,       Yes
6         2,  -4,        -
7         3,  -5,       Yes
8         4,  -6,        -
9         3,  -7,        -
10        2,  -6,        -
11        3,  -5,       Yes
...
Illustration of the nodes of the structure:
-----------------------------------------------------
After 9 stages    After 10 stages    After 11 stages
-----------------------------------------------------
.
.    1                 1                  1
.  0   2             0   2              0   2
.        3                 3                  3
.          4                 4                  4
.        5                 5                  5
.      6                 6                  6
.        7                 7                 11
.          8            10   8             10   8
.        9                 9                  9
.

Links

Crossrefs

Bisections: A211010, A211011.
Cf. A187210, A210838, A210841, A211001-A211003, A211020-A211024, A355478, A357434.

Programs

  • Mathematica
    A211000[nmax_]:=Module[{walk={{0,0}},angle=3/4Pi,turn=Pi/2},Do[If[!PrimeQ[n],If[n>5&&PrimeQ[n-1],turn*=-1];angle-=turn];AppendTo[walk,AngleVector[Last[walk],{Sqrt[2],angle}]],{n,0,nmax-1}];walk];
A211000[100] (* Generates 101 coordinate pairs *) (* Paolo Xausa, Aug 23 2022 *)
  • PARI
    A211000(nmax) = my(walk=vector(nmax+1), turn=1, p1, p2); walk[1]=[0,0];if(nmax==0,return(walk));walk[2]=[1,1];for(n=1, nmax-1, p1=walk[n];p2=walk[n+1];if(isprime(n),walk[n+2]=[2*p2[1]-p1[1],2*p2[2]-p1[2]],if(n>5 && isprime(n-1), turn*=-1);walk[n+2]=[p2[1]-turn*(p1[2]-p2[2]),p2[2]+turn*(p1[1]-p2[1])]));walk;
A211000(100) \\ Generates 101 coordinate pairs - Paolo Xausa, Sep 22 2022
  • Python
    from sympy import isprime
def A211000(nmax):
    walk, turn = [(0,0),(1,1)], 1
    for n in range(1,nmax):
        p1, p2 = walk[-2], walk[-1]
        if isprime(n): # Go straight
            walk.append((2*p2[0]-p1[0],2*p2[1]-p1[1]))
        else:          # Turn
            if n>5 and isprime(n-1): turn *= -1
            walk.append((p2[0]-turn*(p1[1]-p2[1]),p2[1]+turn*(p1[0]-p2[0])))
    return walk[:nmax+1]
print(A211000(100)) # Generates 101 coordinate pairs - Paolo Xausa, Sep 22 2022

A211020 Number of circles in the structure of A211000 after n-th stage.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8
Offset: 0

Views

Author

Omar E. Pol, Mar 30 2012

Keywords

Comments

For n >= 13 the structure looks like essentially a column of tangent circles of radius 1. The circles are centered on a straight line which is parallel to the axis "y". The structure arises from the prime numbers A000040.

Examples

			From _Paolo Xausa_, Jan 09 2023: (Start)
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; circles are depicted as rhombi).
n       0       5      11      13      15      34      41      65
a(n)    0       0       1       2       3       4       5       6
.
                                                                /\
                                                                \/
                                                                 \
                                                                 /
                                                        /\      /\
                                                        \/      \/
              /\      /\      /\      /\      /\/\    /\/\    /\/\
                \       \       \       \       \/      \/      \/
                 \       \       \      /\      /\      /\      /\
                 /       /       /      \/      \/      \/      \/
                        /       /\      /\      /\      /\      /\
                        \       \/      \/      \/      \/      \/
                        /\      /\      /\      /\      /\      /\
                        \/      \/      \/      \/      \/      \/
(End)

Links

Crossrefs

Cf. A187210, A210838, A210841.
Cf. A211000, A211001, A211002, A211003.
Cf. A211010, A211011, A211021, A211022, A211023.
Cf. A355478, A357434.

Programs

  • Mathematica
    A211020[nmax_]:=Module[{ep={{0, 0}}, angle=3/4Pi, turn=Pi/2, cells}, Join[{0}, Table[If[!PrimeQ[n], If[n>5&&PrimeQ[n-1], turn*=-1]; angle-=turn]; AppendTo[ep, AngleVector[Last[ep], {Sqrt[2], angle}]]; cells=FindCycle[Graph[MapApply[UndirectedEdge, Partition[ep, 2, 1]]], {4}, All]; CountDistinct[Map[Sort, Map[First, cells, {2}]]], {n, 0, nmax-1}]]];
A211020[100] (* Paolo Xausa, Jan 06 2023 *)

Extensions

Offset changed to 0 and a(0) prepended by Paolo Xausa, Jan 06 2023
Showing 1-5 of 5 results.

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