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

Author

Paolo Xausa, Aug 06 2023

Keywords

• nonn

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.

Examples

			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-
.
		

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

Crossrefs

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

Programs

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 *)