A180001 -id:A180001 - cp's OEIS frontend

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).

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

N. J. A. Sloane, May 05 2020

nonn

			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.

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

Bert Dobbelaere, C++ program
Bradley Klee, Table giving R, then {s(R,1)..s(R,10)} for 0 <= R <= 255
Index entries for sequences related to cellular automata

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

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

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

N. J. A. Sloane, May 05 2020

nonn

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

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

Bert Dobbelaere, Table of n, a(n) for n = 1..60

Cf. A085587, A180001, A334499-A334515.

Cf. also A334497.

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 *)

More terms from Bert Dobbelaere, May 09 2020

A334515 Eventual period of a single cell in rule 75 cellular automaton in a cyclic universe of width n.

Original entry on oeis.org

2, 2, 3, 2, 30, 18, 126, 2, 504, 430, 979, 18, 443, 2198, 6820, 976, 78812, 7812, 158080, 142580, 248493, 122870, 1630792, 18, 2777040, 4511, 81688176, 868, 463347935, 5921860, 1211061438, 26636800, 598772163, 40012662, 145710075, 135322524, 40583131393, 535150200, 132932362849, 3936823600

Offset: 1

N. J. A. Sloane, May 05 2020

nonn

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

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

Cf. A180001, A334496, A334499-A334514.

a(11)-a(14) from Jinyuan Wang, May 09 2020

More terms from Bert Dobbelaere, May 09 2020

A085587 Eventual period of a single cell in rule 90 cellular automaton in a cyclic universe of width n.

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, 63, 62, 2047, 8, 1023, 126, 511, 28, 16383, 30, 31, 1, 31, 30, 4095, 28, 87381, 1022, 4095, 24, 1023, 126, 127, 124, 4095, 4094, 8388607, 16, 2097151, 2046, 255, 252, 67108863, 1022, 1048575, 56, 511, 32766, 536870911, 60

Offset: 1

N. J. A. Sloane, Jul 03 2003

nonn

O. Martin, A. M. Odlyzko, 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.
S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601-644. (See Section 4.) [Added by _N. J. A. Sloane_, Apr 20 2010]

Cf. A085588, A085589, A085590, A085591, A085592, A085593, A085595.

Cf. also A180001, A334496, A334499-A334515.

More terms from Sean A. Irvine, Jun 10 2018

Definition edited by N. J. A. Sloane, May 05 2020

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

Ben Branman, Jan 30 2011

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

			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.

T. D. Noe, Table of n, a(n) for n = 0..1023
Index entries for sequences related to cellular automata
Eric Weisstein's World of Mathematics, Rule 110
Wikipedia, Rule 110

Cf. A006978, A070887, A071049, A180001, A186083, A204371.

Cf. also A269160, A269174, A269175.

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

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

A204371 Maximum period of cellular automaton rule 110 in a cyclic universe of width n.

Original entry on oeis.org

1, 1, 1, 2, 1, 9, 14, 16, 7, 25, 110, 18, 351, 91, 295, 32, 578, 81, 285, 240, 630, 462, 1058, 552, 300, 351, 567, 2156, 1044, 1770, 2759, 2368, 1100, 969, 3920, 1584

Offset: 1

Ben Branman, Jan 14 2012

a(n) >= A180001(n), and this sequence agrees with A180001 up to n=11.

			The 12 cell pattern
000100110111
001101111101
011111000111
110001001101
010011011111
110111110001
011100010011
110100110111
011101111100
110111000100
111101001101
000111011111
001101110001
011111010011
110001110111
010011011100
110111110100
111100011101
000100110111
Has period 18, which is the maximum possible, so a(12)=18

Eric Weisstein's World of Mathematics, Rule 110
Index entries for sequences related to cellular automata

Cf. A180001, A161903, A006978, A332717, A332718.

f[list_] := -Subtract @@ Flatten[Map[Position[#, #[[-1]]] &, NestWhileList[CellularAutomaton[110], list, Unequal, All], {0}]]; ma[n_] := Max[Table[f[IntegerDigits[i, 2, n]], {i, 0, 2^n - 1}]]; Table[ma[n], {n, 1, 10}]

a(19)-a(36) from Lars Blomberg, Dec 24 2015

A334506 Eventual period of a single cell in rule 161 cellular automaton in a cyclic universe of width n.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 1, 6, 6, 4, 4, 14, 14, 1, 1, 14, 14, 12, 12, 62, 62, 8, 8, 126, 126, 28, 28, 30, 30, 1, 1, 30, 30, 28, 28, 1022, 1022, 24, 24, 126, 126, 124, 124, 4094, 4094, 16, 16, 2046, 2046, 252, 252, 1022, 1022, 56, 56, 32766, 32766, 60, 60, 62, 62, 1, 1, 62, 62, 60, 60, 8190

Offset: 1

N. J. A. Sloane, May 05 2020

nonn

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

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

Angelo Rosso, Table of n, a(n) for n = 1..100

Cf. A180001, A334496, A334499-A334515.

a(11)-a(40) from Jinyuan Wang, May 09 2020
a(41)-a(50) from Vaclav Kotesovec, May 10 2020
a(51) and beyond from Angelo Rosso, Jul 26 2022

A334508 Eventual period of a single cell in rule 45 cellular automaton in a cyclic universe of width n.

Original entry on oeis.org

2, 2, 1, 2, 30, 18, 126, 2, 504, 430, 979, 18, 676, 2198, 2340, 976, 78812, 3756, 183920, 142580, 352884, 122870, 1358104, 56544, 2777040, 4511, 44568603, 304913, 463347935, 5921860, 855372646, 26636800, 3315517623, 2359940, 24752893585, 135322524, 8049125817, 535150200, 132932362849

Offset: 1

N. J. A. Sloane, May 05 2020

nonn

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

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

Bert Dobbelaere, Table of n, a(n) for n = 1..46

Cf. A180001, A334496, A334499-A334515.

a(11)-a(16) from Jinyuan Wang, May 09 2020

More terms from Bert Dobbelaere, May 11 2020

A334497 Maximum value of eventual period for any starting configuration for a rule 30 cellular automaton in a cyclic universe of width n.

Original entry on oeis.org

1, 1, 1, 8, 5, 1, 63, 40, 171, 15, 154, 102, 832, 1428, 1455, 6016, 10846, 2844, 3705, 6150, 2793, 3553, 38249, 185040, 588425, 312156, 240300, 249165, 1466066, 374265, 2841150, 2002272, 2038476, 5656002, 18480630, 2237472

Offset: 1

N. J. A. Sloane, May 05 2020

Bradley Klee computed a(1)-a(7).

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

Dustin Gage, Elizabeth Laub and Briana McGarry, Cellular Automata: Is Rule 30 Random?, 2005.

Cf. A180001, A334496, A334499-A334515, A357950.

a[rule_, init_] := -Subtract @@ Flatten[Map[
     Position[#, #[[-1]]] &, NestWhileList[CellularAutomaton[rule],
      init, Unequal, All], {0}]]
tri[n_] := a[30, #] & /@ Tuples[{0, 1}, n];
tri /@ Range[7]
Max /@ %
(* Bradley Klee, Apr 26 2020 *)

a(n) <= A357950(n). Equality holds for n = 4, 8, 16. - Pontus von Brömssen, Oct 22 2022

a(8)-a(12) from Jinyuan Wang, May 14 2020
a(13)-a(22) from Pontus von Brömssen, Oct 22 2022
a(23)-a(36) from Paolo Xausa, Jun 29 2023, using data from Gage, Laub and McGarry (2005), p. 7, Table 2.

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

N. J. A. Sloane, May 05 2020

nonn

Nearest integer to A334499(n)/n.

			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.

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

Bradley Klee, Table giving R, then {s(R,1)..s(R,10)} for 0 <= R <= 255

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

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

cp's OEIS Frontend

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).

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

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A334515 Eventual period of a single cell in rule 75 cellular automaton in a cyclic universe of width n.

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A085587 Eventual period of a single cell in rule 90 cellular automaton in a cyclic universe of width n.

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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.

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Mathematica

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A204371 Maximum period of cellular automaton rule 110 in a cyclic universe of width n.

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A334506 Eventual period of a single cell in rule 161 cellular automaton in a cyclic universe of width n.

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A334508 Eventual period of a single cell in rule 45 cellular automaton in a cyclic universe of width n.

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A334497 Maximum value of eventual period for any starting configuration for a rule 30 cellular automaton in a cyclic universe of width n.

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