A071044 -id:A071044 - cp's OEIS frontend

A071029 Triangle read by rows giving successive states of cellular automaton generated by "Rule 22".

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

Offset: 0

Hans Havermann, May 26 2002

Row n has length 2n+1.

			From _Michael De Vlieger_, Oct 05 2015: (Start)
First 8 rows, replacing "0" with "." for better visibility of ON cells:
                1
              1 1 1
            1 . . . 1
          1 1 1 . 1 1 1
        1 . . . . . . . 1
      1 1 1 . . . . . 1 1 1
    1 . . . 1 . . . 1 . . . 1
  1 1 1 . 1 1 1 . 1 1 1 . 1 1 1
1 . . . . . . . . . . . . . . . 1
(End)

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

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Illustration of first 256 generations
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata

For number of ON cells see A071044.

clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; clip /@ CellularAutomaton[22, {{1}, 0}, 9] // Flatten (* Michael De Vlieger, Oct 05 2015 *)

Corrected by Hans Havermann, Jan 07 2012

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

cp's OEIS Frontend

A071029 Triangle read by rows giving successive states of cellular automaton generated by "Rule 22".

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