A070887 -id:A070887 - cp's OEIS frontend

A071048 Number of 0's in n-th row of triangle in A070887.

0, 0, 0, 1, 0, 3, 2, 2, 1, 5, 5, 4, 5, 6, 4, 5, 4, 9, 8, 9, 8, 8, 7, 10, 11, 13, 14, 11, 7, 10, 15, 15, 9, 15, 21, 17, 12, 20, 19, 15, 17, 23, 19, 13, 18, 20, 23, 26, 17, 19, 23, 28, 24, 20, 25, 25, 20, 24, 25, 24, 26, 28, 28, 28, 22, 26, 31, 30, 32, 31, 31, 28, 29, 32

Offset: 0

Hans Havermann, May 26 2002

nonn

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Index entries for sequences related to cellular automata

A070886 Triangle read by rows giving successive states of cellular automaton generated by "Rule 90".

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

N. J. A. Sloane, May 19 2002

If either neighbor is 1 then new state is 1, otherwise new state is 0.
Row n has length 2n+1.
Rules #18, #26, #82, #90, #146, #154, #210, #218 all give rise to this sequence. - Hans Havermann

			1; 1,0,1; 1,0,0,0,1; 1,0,1,0,1,0,1; ...

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 25.

Robert Price, Table of n, a(n) for n = 0..9999
Eric Weisstein's World of Mathematics, Rule 90
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata

Cf. A070950, A070887. Alternate rows of A047999. Interpreted as binary numbers: A038183. Interpreted as Zeckendorf-expansions: A048757. Drawn as binary trees: A080263.

rows = 10; ca = CellularAutomaton[90, {{1}, 0}, rows-1]; Flatten[ Table[ca[[k, rows-k+1 ;; rows+k-1]], {k, 1, rows}]] (* Jean-François Alcover, May 24 2012 *)

More terms from Hans Havermann, May 26 2002

A006978 Successive states of the Rule 110 cellular automaton defined by 000, 001, 010, 011, ..., 111 -> 0,1,1,1,0,1,1,0 when started with a single ON cell.

Original entry on oeis.org

1, 3, 7, 13, 31, 49, 115, 215, 509, 775, 1805, 3359, 7985, 12659, 29655, 54909, 130759, 197581, 460383, 855793, 2038675, 3227319, 7562237, 14149127, 33304077, 50625055, 118279729, 220060275, 523730647, 830325757, 1942439431, 3595423245, 8571017759, 12951092785

Offset: 1

N. J. A. Sloane

nonn

Marc LeBrun, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Paolo Xausa, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rule 110
Index entries for sequences related to cellular automata

Cf. A070887, A117999.

Map[FromDigits[#, 2] &, CellularAutomaton[110, {{1}, 0}, 30]] (* Ben Branman, Dec 28 2010 *)

a(n) = A117999(n-1)/2^(n-1). - Pontus von Brömssen, Oct 18 2022

More terms from Eric W. Weisstein, Apr 11 2006

Definition clarified by N. J. A. Sloane, Oct 18 2022

A071049 Number of 1's in n-th generation of 1-D CA using Rule 110, started with a single 1.

Original entry on oeis.org

1, 2, 3, 3, 5, 3, 5, 6, 8, 5, 6, 8, 8, 8, 11, 11, 13, 9, 11, 11, 13, 14, 16, 14, 14, 13, 13, 17, 22, 20, 16, 17, 24, 19, 14, 19, 25, 18, 20, 25, 24, 19, 24, 31, 27, 26, 24, 22, 32, 31, 28, 24, 29, 34, 30, 31, 37, 34, 34, 36, 35, 34, 35, 36, 43, 40, 36, 38, 37, 39, 40

Offset: 0

Hans Havermann, May 26 2002

nonn

Number of 1's in n-th row of triangle in A070887.

Although the initial behavior is chaotic, it is an astonishing fact, pointed out by Wolfram [2002, p. 39], that after about three thousand terms all the irregularities disappear. - N. J. A. Sloane, May 15 2015

Matthew Cook, A Concrete View of Rule 110 Computation, in "The Complexity of Simple Programs", T. Neary, D. Woods, A. K. Seda, and N. Murphy (Eds.), 2008, pp. 31-55.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Vincenzo Librandi and N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (First 1000 terms from Vincenzo Librandi)
Matthew Cook, A Concrete View of Rule 110 Computation, arXiv:0906.3248 [cs.CC], 2009.
Matthew Cook, Universality in Elementary Cellular Automata, Complex Systems 15 (2004), 1-40.
N. J. A. Sloane, Illustration of first 20 generations
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Rule 110
Wikipedia, Rule 110
Index entries for sequences related to cellular automata

Cf. A070950, A070952, A070997, A071029, A071044, A151931, A246027.

A071049 := proc(n)
    add( A070887(n+1,k),k=1..n+1) ;
end proc:
seq(A071049(n),n=0..20) ; # R. J. Mathar, Feb 18 2015

Total /@ CellularAutomaton[110,{{1},0},100] (* N. J. A. Sloane, Aug 10 2009 *)

For n >= 2854, a(n+469) = -a(n+453) + a(n+256) + a(n+240) + a(n+229) + a(n+213) - a(n+16) - a(n). - N. J. A. Sloane, May 15 2015

Added references and links. - N. J. A. Sloane, Aug 09 2014

Changed offset to make consistent with A070952, etc. - N. J. A. Sloane, Aug 15 2014

A075437 Triangle read by rows giving successive iterations of the Rule 110 elementary cellular automaton starting with a single black (1) cell where row n is of length 2n+1.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 0

Eric W. Weisstein, Sep 15 2002

The right n terms in a(n) are all 0.

T(n,k) = A070887(n+1,k+1), k=0..n. - Reinhard Zumkeller, Jun 26 2013

			      1;
    1,1,0;
  1,1,1,0,0;
1,1,0,1,0,0,0; ...

S. Wolfram, A New Kind of Science. Champaign, IL: Wolfram Media, p. 31ff, 2002.

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Rule 110
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A070887.

a075437 n k = a075437_tabf !! n !! k
a075437_row n = a075437_tabf !! n
a075437_tabf = iterate rule110 [1] where
   rule110 row = f ([0,0] ++ row ++ [0,0]) where
       f [,]          = []
       f (:ws@(0:0:)) = 0 : f ws
       f (1:ws@(1:1:_)) = 0 : f ws
       f (:ws@(:_:_)) = 1 : f ws
-- Reinhard Zumkeller, Jun 26 2013

A075437list[rowmax_]:=MapIndexed[ArrayPad[#1,#2-rowmax-1]&,CellularAutomaton[110,{{1},0},{rowmax,All}]];A075437list[10] (* Generates 11 rows *) (* Paolo Xausa, Oct 04 2023 *)

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

A237119 Number of white areas in the graph of elementary cellular automaton with rule 110 at generation n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 17, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 43, 46, 50, 52, 55, 57, 59, 62, 65, 68, 71, 74, 78, 81, 85, 90, 95, 98

Offset: 0

Philippe Beaudoin, Feb 03 2014

nonn

Philippe Beaudoin, Table of n, a(n) for n = 0..999
Philippe Beaudoin, Python program to generate the sequence for rule 30 (can be trivially modified for this sequence).
Eric Weisstein's World of Mathematics, Rule 110.

Cf. A070887 (rule 110), A237118 (for rule 30), A237120 (for rule 150).

cp's OEIS Frontend

A071048 Number of 0's in n-th row of triangle in A070887.

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A070886 Triangle read by rows giving successive states of cellular automaton generated by "Rule 90".

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A006978 Successive states of the Rule 110 cellular automaton defined by 000, 001, 010, 011, ..., 111 -> 0,1,1,1,0,1,1,0 when started with a single ON cell.

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A071049 Number of 1's in n-th generation of 1-D CA using Rule 110, started with a single 1.

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Maple

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A075437 Triangle read by rows giving successive iterations of the Rule 110 elementary cellular automaton starting with a single black (1) cell where row n is of length 2n+1.

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Haskell

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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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Programs

Mathematica

Formula

A237119 Number of white areas in the graph of elementary cellular automaton with rule 110 at generation n.

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