cp's OEIS Frontend

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.

%I A363346 #41 Feb 16 2025 08:34:05
%S A363346 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,
%T A363346 15,18,18,20,19,23,19,28,22,29,30,35,31,36,34,39,36,41,38,47,40,48,48,
%U A363346 50,48,51,50,54,50,55,55,56,56,56,59,56,60,61,63,62,64
%N 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.
%H A363346 Paolo Xausa, <a href="/A363346/b363346.txt">Table of n, a(n) for n = 1..1000</a>
%H A363346 Michael Brunnbauer, <a href="https://brunni.de/findings30/">Diagonals in elementary cellular automaton 30</a>, 2019 (<a href="/A363346/a363346.pdf">local PDF copy</a>, with author's permission).
%H A363346 Eric S. Rowland, <a href="https://wpmedia.wolfram.com/uploads/sites/13/2018/02/16-3-4.pdf">Local Nested Structure in Rule 30</a>, Complex Systems 16 (2006), pp. 239-258.
%H A363346 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule30.html">Rule 30</a>.
%H A363346 Stephen Wolfram, <a href="https://www.wolframscience.com/nks/notes-2-1--rule-30/">Notes on Chapter 2, Rule 30</a>, from A New Kind of Science Online, Wolfram Media, 2002.
%H A363346 Stephen Wolfram, <a href="https://writings.stephenwolfram.com/2019/10/announcing-the-rule-30-prizes/">Announcing the Rule 30 Prizes</a>, Stephen Wolfram Writings, 2019.
%H A363346 Paolo Xausa, <a href="/A363346/a363346_1.png">First 1000 evolution steps</a>, with transient cells in blue shades.
%H A363346 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%e A363346 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).
%e A363346 .
%e A363346                               3rd diagonal
%e A363346                            __ Transient = 1
%e A363346                         - /   Repeat    = 0
%e A363346                        --1    a(3)      = 1
%e A363346                       --0--
%e A363346                      --0----      12th diagonal
%e A363346                     --0------  __ Transient = 01
%e A363346                    --0--------/   Repeat    = 0010
%e A363346                   --0--------0-   a(12)     = 2
%e A363346                  --0--------1---
%e A363346                 --0--------0-----  __ 20th diagonal
%e A363346                --0--------0-------/   Transient = 01000101
%e A363346               --0--------1-------0-   Repeat    = 1100
%e A363346              --0--------0-------1---  a(20)     = 8
%e A363346             --0--------0-------0-----
%e A363346            --0--------0-------0-------
%e A363346           --0--------1-------0---------
%e A363346          --0--------0-------1-----------
%e A363346         --0--------0-------0-------------
%e A363346        --0--------0-------1---------------
%e A363346       --0--------1-------1-----------------
%e A363346      --0--------0-------1-------------------
%e A363346     --0--------0-------0---------------------
%e A363346    --0--------0-------0-----------------------
%e A363346   --0--------1-------1-------------------------
%e A363346 .
%e A363346 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).
%e A363346 .
%e A363346                         -
%e A363346                        --*
%e A363346                       -----
%e A363346                      -------
%e A363346                     ---------
%e A363346                    ----------*
%e A363346                   -----------*-
%e A363346                  -----------*-**
%e A363346                 -------------****
%e A363346                -------------******
%e A363346               --------------*******
%e A363346              ---------------********
%e A363346   Order     ---------------**********     Disorder
%e A363346            ----------------***********
%e A363346           ----------------*************
%e A363346          ----------------*-*************
%e A363346         ------------------***************
%e A363346        ------------------*****************
%e A363346       ------------------*-*****************
%e A363346      ------------------*-*******************
%e A363346     --------------------*********************
%e A363346    --------------------***********************
%e A363346   --------------------*************************
%e A363346 .
%t A363346 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]
%o A363346 (Python) # See Brunnbauer link, Appendix 3.
%Y A363346 Cf. A070950, A094605 (periods of diagonals from the right).
%Y A363346 Cf. A363344 (diagonals), A363345 (eventual periods), A364241.
%K A363346 nonn
%O A363346 1,12
%A A363346 _Paolo Xausa_, May 28 2023