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 1-4 of 4 results.

A161903 Convert n into a sequence of binary digits, apply one step of the rule 110 cellular automaton, and interpret the results as a binary integer.

Original entry on oeis.org

0, 3, 6, 7, 12, 15, 14, 13, 24, 27, 30, 31, 28, 31, 26, 25, 48, 51, 54, 55, 60, 63, 62, 61, 56, 59, 62, 63, 52, 55, 50, 49, 96, 99, 102, 103, 108, 111, 110, 109, 120, 123, 126, 127, 124, 127, 122, 121, 112, 115, 118, 119, 124, 127, 126, 125, 104, 107, 110, 111, 100, 103, 98, 97, 192, 195, 198, 199, 204, 207, 206, 205, 216, 219, 222, 223, 220, 223, 218, 217, 240, 243, 246, 247, 252, 255, 254, 253, 248, 251, 254, 255, 244, 247, 242, 241, 224, 227, 230, 231, 236
Offset: 0

Views

Author

Ben Branman, Jan 30 2011

Keywords

Comments

a(a(a(...1))) (n times) gives A006978(n)

Examples

			For n=19, the evolution after one step is
0, 1, 0, 0, 1, 1  (n=19)
1, 1, 0, 1, 1, 1  (a(n)=55)
So a(n)=55.

Links

Crossrefs

Cf. A006978, A070887, A071049, A180001, A186083, A204371.
Cf. also A269160, A269174, A269175.

Programs

  • Mathematica
    a[n_] :=
FromDigits[
  Drop[Part[CellularAutomaton[110, {IntegerDigits[n, 2], 0}], 1], -1],
   2];Table[a[n],{n,0,100}]

Formula

a(n) = A057889(A269174(A057889(n))). - Antti Karttunen, Jun 02 2018

A332717 Triangle read by rows in which row n lists the possible eventual periods of cellular automaton rule 110 in a cyclic universe of width n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 9, 1, 14, 1, 2, 8, 16, 1, 3, 7, 1, 5, 15, 25, 1, 7, 110, 1, 2, 9, 18, 1, 351, 1, 7, 12, 14, 21, 91, 1, 295, 1, 2, 8, 16, 24, 32, 1, 7, 119, 578, 1, 3, 7, 9, 27, 81, 1, 190, 285, 1, 2, 5, 7, 15, 25, 30, 50, 200, 240, 1, 14, 21, 189, 315, 630
Offset: 1

Views

Author

Hans Havermann, Jun 08 2020

Keywords

Comments

Frequency of occurrence for the first 73 terms (semicolons separate rows): 2; 4; 8; 4, 12; 32; 10, 54; 9, 119; 20, 12, 8, 216; 17, 18, 477; 134, 220, 130, 540; 35, 495, 1518; 34, 12, 3426, 624; 54, 8138; 67, 8442, 644, 371, 168, 6692; 113, 32655; 116, 12, 8, 680, 1920, 62800; 138, 93330, 15895, 21709; 181, 36, 2349, 57024, 198594, 3960; 249, 28481, 495558; 534, 12, 3040, 49700, 270, 3300, 614140, 30660, 342380, 4540; 414, 455, 42, 7938, 331590, 1756713. Note that the sum of the frequencies of row n is 2^n.

Examples

			Triangle begins:
  1
  1
  1
  1    2
  1
  1    9
  1   14
  1    2    8   16
  1    3    7
  1    5   15   25
  1    7  110
  1    2    9   18
  1  351
  1    7   12   14   21   91
  1  295
  1    2    8   16   24   32
  1    7  119  578
  1    3    7    9   27   81
  1  190  285
  1    2    5    7   15   25   30   50  200  240
  1   14   21  189  315  630

Crossrefs

Cf. A332718 (row lengths), A204371 (final terms for each row).

A332718 Number of possible eventual periods of cellular automaton rule 110 in a cyclic universe of width n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 4, 3, 4, 3, 4, 2, 6, 2, 6, 4, 6, 3, 10, 6
Offset: 1

Views

Author

Hans Havermann, Jun 08 2020

Keywords

Crossrefs

Cf. A332717, A204371.

A205599 Maximum period of the totalistic 2-color radius 2 cellular automaton in a cyclic universe of width n.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 14, 4, 22, 2, 121, 5, 143, 14, 55, 26, 17, 22, 171, 180, 189, 198, 207
Offset: 1

Views

Author

Ben Branman, Jan 29 2012

Keywords

Comments

A cell's neighborhood consists of itself, the two cells to its left, and the two cells to its right. A cell becomes live if it had either two or four live neighbors (including itself) in the previous generation.

Examples

			For n=7, the initial state 0, 0, 1, 1, 0, 1, 0 has evolution:
0011010
1110010
1000110
1011100
1010001
0010111
0110100
1100101
0001101
0111001
0100011
0101110
1101000
1001011
0011010
Which has period 14, the highest possible.  Thus a(7)=14.

References

  • Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, 2002, p. 255-260, p. 281-285

Crossrefs

Cf. A204371, A204714.

Programs

  • Mathematica
    f[list_] := -Subtract @@ Flatten[Map[Position[#, #[[-1]]] &, NestWhileList[CellularAutomaton[{20, {2, 1}, 2}], list, Unequal, All], {0}]]; a[n_] := Max[Table[f[IntegerDigits[i, 2, n]], {i, 0, 2^n - 1}]]; Table[a[n], {n, 1, 12}]
Showing 1-4 of 4 results.

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