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 1-10 of 15 results. Next

A010760 Duplicate of A085587.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 7, 1, 7, 6, 31, 4, 63, 14, 15, 1, 15, 14, 511, 12
Offset: 1

Views

Author

gandalf(AT)hrn.office.ssi.net (Jim Ausfahl)

Keywords

A085594 Duplicate of A085587.

Original entry on oeis.org

1, 1, 3, 2, 7, 1, 7, 6, 31, 4, 63, 14, 15, 1, 15, 14, 511, 12, 63, 62, 2047, 8, 1023, 126
Offset: 3

Views

Author

Keywords

A180001 Eventual period of a single cell in rule 110 cellular automaton in a cyclic universe of width n.

Original entry on oeis.org

1, 1, 1, 2, 1, 9, 14, 16, 7, 25, 110, 9, 351, 91, 295, 32, 7, 27, 285, 30, 630, 44, 1058, 36, 250, 7, 405, 1652, 1044, 60, 7, 64, 495, 51, 1050, 72, 4403, 76, 390, 60, 7, 630, 1548, 88, 7, 7, 705, 96, 1470, 100, 765, 195, 8109, 7, 825, 7, 2052, 116, 7, 19560, 915
Offset: 1

Views

Author

Ben Branman, Jan 13 2011

Keywords

Comments

The first 21 terms match the most frequent possible outcome (see comment in A332717) with the exception of a(14) which is the second-most frequent. - Hans Havermann, Jun 11 2020

Examples

			For n=4, the evolution of a single cell is:
0001
0011
0111 <--= period starts
1101
0111 <--= again start of period
etc, so a(4)=2.

Links

Crossrefs

Cf. A085587, A334496, A334497, A332717, A334499-A334515.

Programs

  • Mathematica
    a[n_] := -Subtract @@
   Flatten[Map[Position[#, #[[-1]]] &,
     NestWhileList[CellularAutomaton[110],
      Prepend[Table[0, {n - 1}], 1], Unequal, All], {0}]]
  • Sage
    def A180001(n):
    def bit(x,i): return (x >> i) & 1
    rulemap = dict((tuple(bit(i,k) for k in reversed(range(3))), bit(110,i)) for i in range(8))
    def neighbours(d, i): return tuple(d[k % n] for k in [i-1..i+1])
    v = [0]*n; v[-1] = 1;
    history = [v]
    while True:
        v2 = [rulemap[neighbours(history[-1], i)] for i in range(n)]
        if v2 in history: return len(history)-history.index(v2)
        history.append(v2) # D. S. McNeil, Jan 15 2011

Extensions

More terms from Alois P. Heinz, Jan 14 2011

A334499 For 0 <= R <= 255, let s(R,n) = eventual period of a single cell in a Rule R cellular automaton operating in a cyclic universe of width n; a(n) = max_R s(R,n).

Original entry on oeis.org

2, 2, 6, 8, 30, 18, 126, 40, 504, 430, 979, 102, 819, 2198, 6820, 6016, 78812, 7812, 183920, 142580, 352884, 122870, 1630792, 185040, 2777040, 312156, 81688176, 304913, 463347935, 5921860, 1211061438, 26636800, 3315517623, 40012662, 24752893585, 135322524, 40583131393, 535150200, 132932362849, 3936823600
Offset: 1

Views

Author

N. J. A. Sloane, May 05 2020

Keywords

Examples

			For R = 45, the sequence {s(R,1)..s(R,10)} is 2,2,1,2,30,18,126,2,504,430 (see A334508), and s(45,10) = 430 is the greatest value of any s(R,10), and a(10) = 430.

References

  • Bradley Klee, Posting to Math Fun Mailing List, Apr 26 2020.

Links

Crossrefs

Cf. A085587, A180001, A334496, A334500-A334515, A357950.

Formula

a(n) <= A357950(n). Equality holds for all n <= 35, except n = 12, 13, 23, 24, 25, 26, 28, 34. - Pontus von Brömssen, Nov 09 2022

Extensions

More terms from Bert Dobbelaere, May 09 2020

A334496 Eventual period of a single cell in rule 30 cellular automaton in a cyclic universe of width n.

Original entry on oeis.org

1, 1, 1, 8, 5, 1, 4, 40, 72, 15, 154, 102, 260, 1428, 1455, 6016, 10846, 2844, 247, 3420, 597, 3256, 38249, 185040, 588425, 312156, 240300, 249165, 833808, 374265, 2841150, 842528, 1049268, 5656002, 18480630, 2844, 49276415, 9329228, 961272, 19211080, 51151354, 109603410
Offset: 1

Views

Author

N. J. A. Sloane, May 05 2020

Keywords

Comments

Bradley Klee computed a(1)-a(10).

References

  • Bradley Klee, Posting to Math Fun Mailing List, Apr 26 2020

Links

Crossrefs

Cf. A085587, A180001, A334499-A334515.
Cf. also A334497.

Programs

  • Mathematica
    a[rule_, n_] := -Subtract @@ Flatten[Map[     Position[#, #[[-1]]] &,
     NestWhileList[CellularAutomaton[rule],
      Prepend[Table[0, {n - 1}], 1], Unequal, All], {0}]]
a[30, #] & /@ Range[10]
(* Bradley Klee, Apr 26 2020 *)

Extensions

More terms from Bert Dobbelaere, May 09 2020

A085595 Maximal cycle lengths in a certain class of one-dimensional cellular automata.

Original entry on oeis.org

1, 4, 6, 2, 14, 8, 14, 12, 62, 4, 42, 28, 30, 16, 30, 28, 1022, 24, 126, 124, 4094, 8, 2046, 84, 1022, 56, 32766, 60, 62, 32, 62, 60, 8190, 56, 58254, 2044, 8190, 48
Offset: 3

Views

Author

N. J. A. Sloane, Jul 03 2003

Keywords

References

  • O. Martin, A. M. Odlyzko and S. Wolfram, Algebraic properties of cellular automata, Comm Math. Physics, 93 (1984), pp. 219-258, Reprinted in Theory and Applications of Cellular Automata, S Wolfram, Ed., World Scientific, 1986, pp. 51-90 and in Cellular Automata and Complexity: Collected Papers of Stephen Wolfram, Addison-Wesley, 1994, pp. 71-113 See Table 1.

Crossrefs

Cf. A085587-A085593.

A085588 Eventual period of a single cell in rule 150 cellular automaton in a cyclic universe of width n.

Original entry on oeis.org

1, 2, 3, 1, 7, 4, 7, 6, 31, 2, 21, 14, 15, 8, 15, 14, 511, 12, 63, 62, 2047, 4, 1023, 42, 511, 28, 16383, 30, 31, 16, 31, 30, 4095, 28, 29127, 1022, 4095, 24, 1023, 126, 127, 124, 4095, 4094, 8388607, 8, 2097151, 2046, 255, 84, 67108863, 1022, 1048575, 56, 511, 32766, 536870911, 60, 17043521, 62, 63, 32, 63, 62
Offset: 3

Views

Author

N. J. A. Sloane, Jul 03 2003

Keywords

Comments

From Roman Khrabrov, Aug 17 2024: (Start)
It appears that 2^A007814(n) * (2^A309786(n) - 1) divides a(n). For rule 90, it follows from Lemma 3.5 and Theorem 3.5 from Martin & Odlyzko & Wolfram's paper, and the definition of A309786. Rule 150 appears to have the same behavior (verified for n <= 1000).
The numbers for which a(n) differs from 2^A007814(n) * (2^A309786(n) - 1), are the powers of 2 and the numbers in the form 6*2^k, 13*2^k, 37*2^k, 61*2^k, 67*2^k, 95*2^k and so on (there is no corresponding OEIS sequence).
It seems that in 2D case (totalistic rule 34 on a toroidal grid) the formula 2^A007814(n) * (2^A309786(n) - 1) gives the correct maximum cycle lengths in all cases except powers of 2. Replacing A007814(n) with A091090(n) appears to always provide the correct maximum cycle lengths, even at powers of 2.
Conjecture: a(n) = n only if n belongs to A115770. The inverse does not hold true in general; the first exception is 445. (End)

References

  • O. Martin, A. M. Odlyzko and S. Wolfram, Algebraic properties of cellular automata, Comm Math. Physics, 93 (1984), pp. 219-258, Reprinted in Theory and Applications of Cellular Automata, S Wolfram, Ed., World Scientific, 1986, pp. 51-90 and in Cellular Automata and Complexity: Collected Papers of Stephen Wolfram, Addison-Wesley, 1994, pp. 71-113 See Table 1.

Links

Crossrefs

Cf. A085587-A085595.

Extensions

More terms from Sean A. Irvine, Jun 10 2018
Name clarified by Roman Khrabrov, Aug 17 2024

A085593 Maximal cycle lengths in a certain class of one-dimensional cellular automata.

Original entry on oeis.org

2, 4, 6, 2, 14, 8, 14, 12, 62, 4, 42, 28, 30, 16, 30, 28, 1022, 24, 126, 124, 4094, 8, 2046, 84, 1022, 56, 32766, 60, 62, 32, 62, 60, 8190, 56, 58254, 2044, 8190, 48
Offset: 3

Views

Author

N. J. A. Sloane, Jul 03 2003

Keywords

References

  • O. Martin, A. M. Odlyzko and S. Wolfram, Algebraic properties of cellular automata, Comm Math. Physics, 93 (1984), pp. 219-258, Reprinted in Theory and Applications of Cellular Automata, S Wolfram, Ed., World Scientific, 1986, pp. 51-90 and in Cellular Automata and Complexity: Collected Papers of Stephen Wolfram, Addison-Wesley, 1994, pp. 71-113 See Table 1.

Crossrefs

Cf. A085587-A085595.

A334500 For 0 <= R <= 255, let s(R,n) = eventual period of a single cell in a Rule R cellular automaton operating in a cyclic universe of width n; a(n) is the nearest integer to max_R s(R,n)/n (rounded down in case of ties).

Original entry on oeis.org

2, 1, 2, 2, 6, 3, 18, 5, 56, 43, 89, 8, 63, 157, 455, 376, 4636, 434, 9680, 7129, 16804, 5585, 70904, 7710, 111082, 12006, 3025488, 10890, 15977515, 197395, 39066498, 832400, 100470231, 1176843, 707225531, 3758959, 1096841389, 14082900, 3408522124, 98420590
Offset: 1

Views

Author

N. J. A. Sloane, May 05 2020

Keywords

Comments

Nearest integer to A334499(n)/n.

Examples

			For R = 45, the sequence {s(R,1)..s(R,10)} is 2,2,1,2,30,18,126,2,504,430 (see A334508), and s(45,10) = 430 is the greatest value of any s(R,10), so a(10) = 430/10 = 430.

References

  • Bradley Klee, Posting to Math Fun Mailing List, Apr 26 2020

Links

Crossrefs

Cf. A085587, A180001, A334496, A334499-A334515, A357867.

Extensions

a(11)-a(40) (based on data in A334499) from Pontus von Brömssen, Oct 15 2022

A085589 Maximal cycle lengths in a certain class of one-dimensional cellular automata.

Original entry on oeis.org

6, 2, 8, 6, 26, 4, 18, 8, 242, 6, 26, 26, 24, 16, 1640, 18, 19682, 16, 78, 242, 177146, 12, 59048, 26, 54, 26, 4782968, 24, 1103762, 160, 726, 1640, 265720, 18, 19682, 19682, 78, 80, 80, 78, 804642554, 484, 72, 177146, 94143178826, 48, 10460353202, 59048, 4920, 52
Offset: 3

Views

Author

N. J. A. Sloane, Jul 03 2003

Keywords

References

  • O. Martin, A. M. Odlyzko and S. Wolfram, Algebraic properties of cellular automata, Comm Math. Physics, 93 (1984), pp. 219-258, Reprinted in Theory and Applications of Cellular Automata, S Wolfram, Ed., World Scientific, 1986, pp. 51-90 and in Cellular Automata and Complexity: Collected Papers of Stephen Wolfram, Addison-Wesley, 1994, pp. 71-113 See Table 1.

Crossrefs

Cf. A085587-A085595.

Extensions

More terms from Sean A. Irvine, Jun 15 2018
Showing 1-10 of 15 results. Next

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

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