A265380 Binary representation of the middle column of the "Rule 158" elementary cellular automaton starting with a single ON (black) cell.

1, 11, 111, 1110, 11101, 111011, 1110111, 11101110, 111011101, 1110111011, 11101110111, 111011101110, 1110111011101, 11101110111011, 111011101110111, 1110111011101110, 11101110111011101, 111011101110111011, 1110111011101110111, 11101110111011101110

Offset: 0

Author

Robert Price, Dec 07 2015

Keywords

• nonn
• easy

Examples

			From _Michael De Vlieger_, Dec 09 2015: (Start)
First 8 rows at left, ignoring "0" outside of range of 1's, the center column values in parentheses, and at right the value of center column cells up to that row :
                        (1)                          -> 1
                      1 (1) 1                        -> 11
                    1 1 (1) 0 1                      -> 111
                  1 1 1 (0) 0 1 1                    -> 1110
                1 1 1 0 (1) 1 1 0 1                  -> 11101
              1 1 1 0 0 (1) 1 0 0 1 1                -> 111011
            1 1 1 0 1 1 (1) 0 1 1 1 0 1              -> 1110111
          1 1 1 0 0 1 1 (0) 0 1 1 0 0 1 1            -> 11101110
        1 1 1 0 1 1 1 0 (1) 1 1 0 1 1 1 0 1          -> 111011101
      1 1 1 0 0 1 1 0 0 (1) 1 0 0 1 1 0 0 1 1        -> 1110111011
    1 1 1 0 1 1 1 0 1 1 (1) 0 1 1 1 0 1 1 1 0 1      -> 11101110111
  1 1 1 0 0 1 1 0 0 1 1 (0) 0 1 1 0 0 1 1 0 0 1 1    -> 111011101110
1 1 1 0 1 1 1 0 1 1 1 0 (1) 1 1 0 1 1 1 0 1 1 1 0 1  -> 1110111011101
(End)
		

References

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

Links

• Robert Price, Table of n, a(n) for n = 0..999
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A071037, A265381.

Programs

Mathematica

f[n_] := Block[{w = {}}, Do[AppendTo[w, Boole[Mod[k, 4] != 3]], {k, 0, n}]; FromDigits@ w]; Table[f@ n, {n, 0, 19}] (* Michael De Vlieger, Dec 09 2015 *)

Formula

Conjectures from Colin Barker, Dec 14 2015 and Apr 18 2019: (Start)

a(n) = 10*a(n-1) + a(n-4) - 10*a(n-5) for n>4.

G.f.: (1+x+x^2) / ((1-x)*(1+x)*(1-10*x)*(1+x^2)).

(End)

Extensions

Removed an unjustified claim that Colin Barker's conjectures are correct. Removed 2 programs based on conjectures. - N. J. A. Sloane, Jun 13 2022