A071029 -id:A071029 - cp's OEIS frontend

A071043 Number of 0's in n-th row of triangle in A071029.

0, 0, 3, 1, 7, 5, 9, 3, 15, 13, 17, 11, 21, 15, 21, 7, 31, 29, 33, 27, 37, 31, 37, 23, 45, 39, 45, 31, 49, 35, 45, 15, 63, 61, 65, 59, 69, 63, 69, 55, 77, 71, 77, 63, 81, 67, 77, 47, 93, 87, 93, 79, 97, 83, 93, 63, 105, 91, 101, 71, 105, 75, 93, 31, 127, 125, 129

Offset: 0

Hans Havermann, May 26 2002

nonn

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

Robert Price, Table of n, a(n) for n = 0..1000
Index entries for sequences related to cellular automata

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

A071044 Number of ON cells at generation n of 1-D CA defined by Rule 22, starting with a single ON cell.

Original entry on oeis.org

1, 3, 2, 6, 2, 6, 4, 12, 2, 6, 4, 12, 4, 12, 8, 24, 2, 6, 4, 12, 4, 12, 8, 24, 4, 12, 8, 24, 8, 24, 16, 48, 2, 6, 4, 12, 4, 12, 8, 24, 4, 12, 8, 24, 8, 24, 16, 48, 4, 12, 8, 24, 8, 24, 16, 48, 8, 24, 16, 48, 16, 48, 32, 96, 2, 6, 4, 12, 4, 12, 8, 24, 4, 12, 8, 24, 8, 24, 16, 48

Offset: 0

Hans Havermann, May 26 2002

nonn

Number of 1's in n-th row of triangle in A071029.

			From _Michael De Vlieger_, Oct 05 2015: (Start)
First 8 rows, replacing "0" with "." for better visibility of ON cells, total of ON cells in each row to the left of the diagram:
1                  1
3                1 1 1
2              1 . . . 1
6            1 1 1 . 1 1 1
2          1 . . . . . . . 1
6        1 1 1 . . . . . 1 1 1
4      1 . . . 1 . . . 1 . . . 1
12   1 1 1 . 1 1 1 . 1 1 1 . 1 1 1
2  1 . . . . . . . . . . . . . . . 1
(End)

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

Robert Price, Table of n, a(n) for n = 0..1000
Kari Eloranta, Partially permutive cellular automata, Nonlinearity 6.6 (1993): 1009. (Further information about Rule 22)
Peter Grassberger, Long-range effects in an elementary cellular automaton, Journal of Statistical Physics, 45.1-2 (1986): 27-39. (Further information about Rule 22)
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata...
N. J. A. Sloane, Illustration of first 21 generations
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601--644.
Index entries for sequences related to cellular automata

Cf. A071029.

ArrayPlot[CellularAutomaton[22, {{1}, 0}, 20]] (* N. J. A. Sloane, Aug 15 2014 *)
Total /@ CellularAutomaton[22, {{1}, 0}, 80] (* Michael De Vlieger, Oct 05 2015 *)

If the binary expansion of n is b_{r-1} b_{r-2} ... b_2 b_1 b_0, then a(n) = 3^b_0 * Prod_{i=1..r-1} 2^b_i = 2^wt(n) if n is even, or (3/2)*2^wt(n) if n is odd (cf. A000120). - N. J. A. Sloane, Aug 09 2014

G.f. = (1+3*x)*Prod_{k >= 1} (1+2*x^(2^k)). - N. J. A. Sloane, Aug 09 2014

Better description from N. J. A. Sloane, Aug 15 2014

A292686 Sierpinski-type iteration: start with a(0)=1, at each step, replace 0 with 000 and 1 with 101.

Original entry on oeis.org

1, 101, 101000101, 101000101000000000101000101, 101000101000000000101000101000000000000000000000000000101000101000000000101000101

Offset: 0

M. F. Hasler, Oct 20 2017

nonn

See A292687 for the decimal representation of a(n) viewed as a "binary number", i.e., as written in base 2.
The Sierpinski carpet (A153490) can be seen as 2-dimensional version of this 1-dimensional variant. The classical Sierpinski gasket triangle (Pascal's triangle mod 2) and "Rule 18" (or Rule 90, A070886) and "Rule 22" (A071029) have similar graphs.
The n-th term a(n) has 3^n digits, the middle third of which are all zero. The digits of a(n) are again the first and last 3^n digits of a(n+1), separated by 3^n zeros.

			a(0) = 1 -> 101 = a(1);
a(1) = 101 -> concat(101,000,101) = 101000101 = a(2).

Michael Coons and James Evans, A sequential view of self--similar measures, or, What the ghosts of Mahler and Cantor can teach us about dimension, arXiv:2011.10722 [math.NT], 2020. See Figure 2 p. 2.

Cf. A292687 for the decimal representation of a(n) viewed as a "binary number", i.e., as written in base 2.
Cf. A153490 (Sierpinski carpet), A047999 (Sierpinski gasket = Pascal's triangle mod 2), A070886 (Rule 18 / Rule 90), A071029 (Rule 22).
Cf. A088917.

A292686[nmax_]:=FoldList[Times,1,100^(3^Range[0,nmax-1])+1];A292686[5] (* Paolo Xausa, May 13 2023 *)

a(n,a=1)=for(k=1,n,a=fromdigits(binary(a)*5,8));fromdigits(binary(a),10) \\ Illustration of the first formula.

PARI
```
A292686(n)=prod(k=0,n-1,100^(3^k)+1)
```

a(n+1) = convert(5*a(n), from base 8, to base 2).
a(n+1) = (100^(3^n)+1)*a(n).
a(n) = Product_{k=0 .. n-1} (100^(3^k)+1).

A266382 Decimal representation of the n-th iteration of the "Rule 22" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 7, 17, 119, 257, 1799, 4369, 30583, 65537, 458759, 1114129, 7798903, 16843009, 117901063, 286331153, 2004318071, 4294967297, 30064771079, 73014444049, 511101108343, 1103806595329, 7726646167303, 18764712120593, 131352984844151, 281479271743489

Offset: 0

Robert Price, Dec 28 2015

Empirical observation: This sequence can also be generated in Conway's Game of Life by setting the initial condition to be an infinite row of ON cells in both directions. After each iteration, rows of cells turned ON will be labeled as 1 and rows that are OFF will be labeled 0. When the resulting binary number is converted to decimal notation, the resulting sequence is the same as A266382. [Brook Estifanos, Mar 09 2016]

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata..., Fig. 6
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A071029.

rule=22; rows=30; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}] (* Decimal Representation of Rows *)

A266381 Binary representation of the n-th iteration of the "Rule 22" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 111, 10001, 1110111, 100000001, 11100000111, 1000100010001, 111011101110111, 10000000000000001, 1110000000000000111, 100010000000000010001, 11101110000000001110111, 1000000010000000100000001, 111000001110000011100000111, 10001000100010001000100010001

Offset: 0

Robert Price, Dec 28 2015

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A071029.

rule=22; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]]],{k,1,rows}] (* Binary Representation of Rows *)

A266383 Total number of ON (black) cells after n iterations of the "Rule 22" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 4, 6, 12, 14, 20, 24, 36, 38, 44, 48, 60, 64, 76, 84, 108, 110, 116, 120, 132, 136, 148, 156, 180, 184, 196, 204, 228, 236, 260, 276, 324, 326, 332, 336, 348, 352, 364, 372, 396, 400, 412, 420, 444, 452, 476, 492, 540, 544, 556, 564, 588, 596, 620, 636

Offset: 0

Robert Price, Dec 28 2015

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A071029.

rule=22; rows=60; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) nbc=Table[Total[catri[[k]]],{k,1,rows}]; (* Number of Black cells in stage n *) Table[Total[Take[nbc,k]],{k,1,rows}] (* Number of Black cells through stage n *)

A266384 Total number of OFF (white) cells after n iterations of the "Rule 22" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 0, 3, 4, 11, 16, 25, 28, 43, 56, 73, 84, 105, 120, 141, 148, 179, 208, 241, 268, 305, 336, 373, 396, 441, 480, 525, 556, 605, 640, 685, 700, 763, 824, 889, 948, 1017, 1080, 1149, 1204, 1281, 1352, 1429, 1492, 1573, 1640, 1717, 1764, 1857, 1944, 2037, 2116

Offset: 0

Robert Price, Dec 28 2015

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A071029.

rule = 22; rows = 60; ca = CellularAutomaton[rule, {{1}, 0}, rows - 1, {All, All}]; (* Start with single black cell *) catri = Table[Take[ca[[k]], {rows - k + 1, rows + k - 1}], {k, 1, rows}]; (* Truncated list of each row *) nbc =
Table[Total[catri[[k]]], {k, 1, rows}]; (* Number of Black cells in stage n *) nwc = Table[Length[catri[[k]]] - nbc[[k]], {k, 1, rows}]; (* Number of White cells in stage n *) Table[Total[Take[nwc, k]], {k, 1, rows}] (* Number of White cells through stage n *)

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A071043 Number of 0's in n-th row of triangle in A071029.

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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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A071044 Number of ON cells at generation n of 1-D CA defined by Rule 22, starting with a single ON cell.

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A292686 Sierpinski-type iteration: start with a(0)=1, at each step, replace 0 with 000 and 1 with 101.

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PARI

PARI

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A266382 Decimal representation of the n-th iteration of the "Rule 22" elementary cellular automaton starting with a single ON (black) cell.

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Mathematica

A266381 Binary representation of the n-th iteration of the "Rule 22" elementary cellular automaton starting with a single ON (black) cell.

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Mathematica

A266383 Total number of ON (black) cells after n iterations of the "Rule 22" elementary cellular automaton starting with a single ON (black) cell.

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Keywords

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Crossrefs

Programs

Mathematica

A266384 Total number of OFF (white) cells after n iterations of the "Rule 22" elementary cellular automaton starting with a single ON (black) cell.

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