A364773 a(n) is the periodic part on the n-th diagonal from the right of rule-30 1-D cellular automaton, when started from a single ON cell.

1, 10, 10, 1100, 10110100, 10101000, 1010011101011000, 11001010101011110011010101010000, 10111010011010101101010101010000, 1010110010110101010110011001111101010011010010101010011001100000, 1010101110101100101010010110101011010010101101010110010110100000

Offset

1,2

Comments

As found by Brunnbauer (2019), if a period doubling occurs at n, then a(n) is of the form AB, where B is the inverse of A. Additionally, the number of trailing zeros of a(n) increases by one when n is even.

Links

Paolo Xausa, Table of n, a(n) for n = 1..25

Michael Brunnbauer, Diagonals in elementary cellular automaton 30, 2019 (local PDF copy, with author's permission).

Eric S. Rowland, Local Nested Structure in Rule 30, Complex Systems 16 (2006), pp. 239-258.

Eric Weisstein's World of Mathematics, Rule 30.

Stephen Wolfram, Notes on chapter 2, Rule 30, from A new kind of science online, Wolfram Media, 2002.

Index entries for sequences related to cellular automata

Example

In the following diagram, showing the first 20 evolution steps of the CA, two diagonals are highlighted (the rest of the CA is represented by hyphens, for better visualization).
.
                2nd diagonal
                   a(2) = 10 __
                               \-
             7th diagonal __   -1-
  a(7) = 1010011101011000   \ ---0-
                             1----1-
                            --0----0-
                           ----1----1-
                          ------0----0-
                         --------0----1-
                        ----------1----0-
                       ------------1----1-
                      --------------1----0-
                     ----------------0----1-
                    ------------------1----0-
                   --------------------0----1-
                  ----------------------1----0-
                 ------------------------1----1-
                --------------------------0----0-
               ----------------------------0----1-
              ------------------------------0----0-
             --------------------------------1----1-
            ----------------------------------0----0-
.

Mathematica

A364773list[steps_]:=Module[{d=2Ceiling[Log2[steps]], ca, n=1, p, plen, a={1}}, ca=CellularAutomaton[30, {{1}, 0}, {steps, {1-d, steps}}]; While[++n<=2(d-1)&&(plen=Length[p=FindRepeat[Flatten[Rest[Split[Diagonal[ca, d-n]]]]]])>=IntegerLength[Last[a]]&&IntegerQ[Log2[plen]], AppendTo[a, FromDigits[p]]]; a];
A364773list[80] (* Analyzes 80 evolution steps *)

Crossrefs

Cf. A070950, A094605 (periods), A363343 (diagonals), A364774 (base 10).

Sequence in context: A279992 A287949 A287627 * A052983 A072804 A079793

Adjacent sequences: A364770 A364771 A364772 * A364774 A364775 A364776

Keyword

nonn

Author

Paolo Xausa, Aug 06 2023

Status

approved