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 1-6 of 6 results.

A002933 Erroneous version of A174313.

Original entry on oeis.org

1, 6, 18, 54, 162, 474, 1398, 4074, 11898, 34554, 100302, 290334, 839466
Offset: 0

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Author

Keywords

References

  • M. E. Fisher and B. J. Hiley, Configuration and free energy of a polymer molecule with solvent interaction, J. Chem. Phys., 34 (1961), 1253-1267.

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.

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, 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

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Author

Paolo Xausa, Jul 18 2022

Keywords

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

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 *)

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]

A038729 Configurations of linear chains in a 6-dimensional hypercubic lattice.

Original entry on oeis.org

12, 132, 1332, 13452, 134892, 1353732, 13536612, 135457932, 1352852292, 13517235732, 134908128732, 1346796414252, 13435850843172
Offset: 1

Views

Author

N. J. A. Sloane, May 02 2000

Keywords

Comments

In the notation of Nemirovsky et al. (1992), a(n), the n-th term of the current sequence is C_{n,m} with m=0 (and d=6). Here, for a d-dimensional hypercubic lattice, C_{n,m} is "the number of configurations of an n-bond self-avoiding chain with m neighbor contacts." (For d=2, we have C(n,0) = A173380(n); for d=3, we have C(n,0) = A174319(n); for d=4, we have C(n,0) = A034006(n); and for d=5, we have C(n,0) = A038726(n).) - Petros Hadjicostas, Jan 03 2019

Links

Crossrefs

Cf. A002932, A002934, A034006, A038726, A173380, A174313, A174319.

Extensions

Terms a(10) and a(11) were copied from Table 1 (p. 1090) in the paper by Nemirovsky et al. (1992) by Petros Hadjicostas, Jan 03 2019
Name edited by Petros Hadjicostas, Jan 03 2019
a(12)-a(13) from Sean A. Irvine, Feb 01 2021

A336758 Number of n-step self-avoiding walks on the honeycomb lattice with no non-contiguous adjacencies.

Original entry on oeis.org

1, 3, 6, 12, 24, 42, 78, 144, 264, 486, 894, 1620, 2964, 5376, 9798, 17760, 32292, 58398, 105960, 191466, 346854, 626172, 1132800, 2043246, 3692406, 6655068, 12015126, 21641526, 39039810, 70277016, 126682584, 227928780, 410605008, 738423492, 1329477732
Offset: 0

Views

Author

Sean A. Irvine, Aug 03 2020

Keywords

Crossrefs

Cf. A001668 (allowing adjacencies), A174313 (hexagonal lattice), A173380 (square lattice).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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