A363346 a(n) is the length of the initial transient, before the periodic part, on the n-th diagonal from the left of rule-30 1-D cellular automaton, when started from a single ON cell.

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 3, 4, 5, 8, 6, 11, 8, 13, 12, 14, 12, 15, 15, 16, 16, 17, 15, 18, 18, 20, 19, 23, 19, 28, 22, 29, 30, 35, 31, 36, 34, 39, 36, 41, 38, 47, 40, 48, 48, 50, 48, 51, 50, 54, 50, 55, 55, 56, 56, 56, 59, 56, 60, 61, 63, 62, 64

Offset

1,12

Links

Paolo Xausa, Table of n, a(n) for n = 1..1000

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.

Stephen Wolfram, Announcing the Rule 30 Prizes, Stephen Wolfram Writings, 2019.

Paolo Xausa, First 1000 evolution steps, with transient cells in blue shades.

Index entries for sequences related to cellular automata

Example

In the following diagram, showing the first 22 evolution steps of the CA, three diagonals are highlighted, along with their transient and periodic parts (the rest of the CA is represented by hyphens, for better visualization).
.
                              3rd diagonal
                           __ Transient = 1
                        - /   Repeat    = 0
                       --1    a(3)      = 1
                      --0--
                     --0----      12th diagonal
                    --0------  __ Transient = 01
                   --0--------/   Repeat    = 0010
                  --0--------0-   a(12)     = 2
                 --0--------1---
                --0--------0-----  __ 20th diagonal
               --0--------0-------/   Transient = 01000101
              --0--------1-------0-   Repeat    = 1100
             --0--------0-------1---  a(20)     = 8
            --0--------0-------0-----
           --0--------0-------0-------
          --0--------1-------0---------
         --0--------0-------1-----------
        --0--------0-------0-------------
       --0--------0-------1---------------
      --0--------1-------1-----------------
     --0--------0-------1-------------------
    --0--------0-------0---------------------
   --0--------0-------0-----------------------
  --0--------1-------1-------------------------
.
In the following diagram the transient cells on every diagonal are represented by asterisks. This results in the division of the CA into two regions: ordered behavior on the left, and apparently chaotic behavior on the right. The boundary between the two regions moves to the left, on average, by about 0.252 cells every evolution step (see Wolfram, 2002 and 2019).
.
                        -
                       --*
                      -----
                     -------
                    ---------
                   ----------*
                  -----------*-
                 -----------*-**
                -------------****
               -------------******
              --------------*******
             ---------------********
  Order     ---------------**********     Disorder
           ----------------***********
          ----------------*************
         ----------------*-*************
        ------------------***************
       ------------------*****************
      ------------------*-*****************
     ------------------*-*******************
    --------------------*********************
   --------------------***********************
  --------------------*************************
.

Mathematica

A363346list[nmax_]:=With[{ca=CellularAutomaton[86, {{1}, 0}, {2nmax, {1-nmax, nmax}}]}, Array[Length[First[FindTransientRepeat[Drop[Diagonal[ca, nmax-#], Ceiling[(#-1)/2]], 2]]]&, nmax]]; A363346list[100]

Prog

(Python) # See Brunnbauer link, Appendix 3.

Crossrefs

Cf. A070950, A094605 (periods of diagonals from the right).

Cf. A363344 (diagonals), A363345 (eventual periods), A364241.

Sequence in context: A109921 A139637 A307734 * A243202 A257136 A344059

Adjacent sequences: A363343 A363344 A363345 * A363347 A363348 A363349

Keyword

nonn

Author

Paolo Xausa, May 28 2023

Status

approved