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

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

Author

Hans Havermann, May 26 2002

Keywords

• nonn

Comments

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

References

• 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.

Links

• 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

Crossrefs

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

Programs

Maple

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

Mathematica

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

Formula

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

Extensions

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