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A364241 a(n) is the initial transient (converted to base 10), 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, or -1 if there is no transient part.

%I A364241 #13 Feb 16 2025 08:34:06
%S A364241 -1,-1,1,-1,-1,-1,-1,-1,-1,-1,1,1,-1,4,9,0,228,63,1241,69,7609,2944,
%T A364241 11356,255,28487,30890,42037,24160,104333,19167,132196,25361,1042145,
%U A364241 473564,4512243,153187,258856349,2950249,353554884,104435283,26762321451,2005002052
%N A364241 a(n) is the initial transient (converted to base 10), 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, or -1 if there is no transient part.
%C A364241 See A363346 for the length of the transients.
%H A364241 Paolo Xausa, <a href="/A364241/b364241.txt">Table of n, a(n) for n = 1..1000</a>
%H A364241 Michael Brunnbauer, <a href="https://brunni.de/findings30/">Diagonals in elementary cellular automaton 30</a>, 2019 (<a href="/A364241/a364241.pdf">local PDF copy</a>, with author's permission).
%H A364241 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 A364241 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule30.html">Rule 30</a>.
%H A364241 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 A364241 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%e A364241 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 A364241 .
%e A364241                               3rd diagonal
%e A364241                            __ Transient = 1
%e A364241                         - /   Repeat    = 0
%e A364241                        --1    a(3)      = 1
%e A364241                       --0--
%e A364241                      --0----      12th diagonal
%e A364241                     --0------  __ Transient = 01
%e A364241                    --0--------/   Repeat    = 0010
%e A364241                   --0--------0-   a(12)     = 1
%e A364241                  --0--------1---
%e A364241                 --0--------0-----  __ 20th diagonal
%e A364241                --0--------0-------/   Transient = 01000101
%e A364241               --0--------1-------0-   Repeat    = 1100
%e A364241              --0--------0-------1---  a(20)     = 69
%e A364241             --0--------0-------0-----
%e A364241            --0--------0-------0-------
%e A364241           --0--------1-------0---------
%e A364241          --0--------0-------1-----------
%e A364241         --0--------0-------0-------------
%e A364241        --0--------0-------1---------------
%e A364241       --0--------1-------1-----------------
%e A364241      --0--------0-------1-------------------
%e A364241     --0--------0-------0---------------------
%e A364241    --0--------0-------0-----------------------
%e A364241   --0--------1-------1-------------------------
%e A364241 .
%t A364241 A364241list[nmax_]:=With[{ca=CellularAutomaton[86,{{1},0},{2nmax,{1-nmax,nmax}}]},Map[If[#=={},-1,FromDigits[#,2]]&,Array[First[FindTransientRepeat[Drop[Diagonal[ca,nmax-#],Ceiling[(#-1)/2]],2]]&,nmax]]];A364241list[50]
%Y A364241 Cf. A070950.
%Y A364241 Cf. A363344 (diagonals), A363345 (periods), A363346 (length of transients), A364240 (periodic part).
%K A364241 sign,base
%O A364241 1,14
%A A364241 _Paolo Xausa_, Jul 15 2023