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A364240 a(n) is the periodic part (converted to base 10), on the n-th diagonal from the left of rule-30 1-D cellular automaton, when started from a single ON cell.

%I A364240 #15 Feb 16 2025 08:34:06
%S A364240 1,1,0,1,1,2,2,0,12,1,6,2,13,11,6,3,11,8,6,12,6,7,3,2,8,9,14,13,0,180,
%T A364240 1,90,74,25,156,224,98,250,123,12,20,239,215,138,20,12,159,192,58,253,
%U A364240 139,86,195,164,25,28,48,235,159,162,213,152,25,8,14,239
%N A364240 a(n) is the periodic part (converted to base 10), on the n-th diagonal from the left of rule-30 1-D cellular automaton, when started from a single ON cell.
%C A364240 See A363345 for the periods.
%H A364240 Paolo Xausa, <a href="/A364240/b364240.txt">Table of n, a(n) for n = 1..1000</a>
%H A364240 Michael Brunnbauer, <a href="https://brunni.de/findings30/">Diagonals in elementary cellular automaton 30</a>, 2019 (<a href="/A364240/a364240.pdf">local PDF copy</a>, with author's permission).
%H A364240 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 A364240 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule30.html">Rule 30</a>.
%H A364240 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 A364240 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%e A364240 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 A364240 .
%e A364240                               3rd diagonal
%e A364240                            __ Transient = 1
%e A364240                         - /   Repeat    = 0
%e A364240                        --1    a(3)      = 0
%e A364240                       --0--
%e A364240                      --0----      12th diagonal
%e A364240                     --0------  __ Transient = 01
%e A364240                    --0--------/   Repeat    = 0010
%e A364240                   --0--------0-   a(12)     = 2
%e A364240                  --0--------1---
%e A364240                 --0--------0-----  __ 20th diagonal
%e A364240                --0--------0-------/   Transient = 01000101
%e A364240               --0--------1-------0-   Repeat    = 1100
%e A364240              --0--------0-------1---  a(20)     = 12
%e A364240             --0--------0-------0-----
%e A364240            --0--------0-------0-------
%e A364240           --0--------1-------0---------
%e A364240          --0--------0-------1-----------
%e A364240         --0--------0-------0-------------
%e A364240        --0--------0-------1---------------
%e A364240       --0--------1-------1-----------------
%e A364240      --0--------0-------1-------------------
%e A364240     --0--------0-------0---------------------
%e A364240    --0--------0-------0-----------------------
%e A364240   --0--------1-------1-------------------------
%e A364240 .
%t A364240 A364240list[nmax_]:=With[{ca=CellularAutomaton[86,{{1},0},{2nmax,{1-nmax,nmax}}]},Array[FromDigits[Last[FindTransientRepeat[Drop[Diagonal[ca,nmax-#],Ceiling[(#-1)/2]],2]],2]&,nmax]];A364240list[100]
%Y A364240 Cf. A070950.
%Y A364240 Cf. A363344 (diagonals), A363345 (periods), A363346 (length of transients), A364241 (transients).
%K A364240 nonn,base
%O A364240 1,6
%A A364240 _Paolo Xausa_, Jul 15 2023