This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A363345 #41 Feb 16 2025 08:34:05
%S A363345 1,1,1,2,1,2,2,1,4,1,4,4,4,4,4,4,4,4,4,4,4,4,4,2,4,4,4,4,1,8,1,8,8,8,
%T A363345 8,8,8,8,8,8,8,8,8,8,8,4,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,4,8,4,8,
%U A363345 8,8,8,8,8,8,8,8,8,8,8,8,4,8,8,8,8,8
%N A363345 a(n) is the eventual period of the n-th diagonal from the left of rule-30 1-D cellular automaton, when started from a single ON cell.
%C A363345 Diagonals from the left of rule-30 cellular automaton are eventually periodic, after an initial transient phase (A363346).
%C A363345 As noted by Wolfram (2002), "Each period doubling turns out to occur exactly when a diagonal in the pattern eventually becomes a white stripe, and the diagonal to its left has an odd number of black cells in each repeating block." See Rowland (2006), p. 256 for a proof of this observation.
%H A363345 Paolo Xausa, <a href="/A363345/b363345.txt">Table of n, a(n) for n = 1..10000</a>
%H A363345 Michael Brunnbauer, <a href="https://brunni.de/findings30/">Diagonals in elementary cellular automaton 30</a>, 2019 (<a href="/A363345/a363345.pdf">local PDF copy</a>, with author's permission).
%H A363345 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 A363345 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule30.html">Rule 30</a>.
%H A363345 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 A363345 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%e A363345 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 A363345 .
%e A363345 3rd diagonal
%e A363345 __ Transient = 1
%e A363345 - / Repeat = 0
%e A363345 --1 a(3) = 1
%e A363345 --0--
%e A363345 --0---- 12th diagonal
%e A363345 --0------ __ Transient = 01
%e A363345 --0--------/ Repeat = 0010
%e A363345 --0--------0- a(12) = 4
%e A363345 --0--------1---
%e A363345 --0--------0----- __ 20th diagonal
%e A363345 --0--------0-------/ Transient = 01000101
%e A363345 --0--------1-------0- Repeat = 1100
%e A363345 --0--------0-------1--- a(20) = 4
%e A363345 --0--------0-------0-----
%e A363345 --0--------0-------0-------
%e A363345 --0--------1-------0---------
%e A363345 --0--------0-------1-----------
%e A363345 --0--------0-------0-------------
%e A363345 --0--------0-------1---------------
%e A363345 --0--------1-------1-----------------
%e A363345 --0--------0-------1-------------------
%e A363345 --0--------0-------0---------------------
%e A363345 --0--------0-------0-----------------------
%e A363345 --0--------1-------1-------------------------
%e A363345 .
%t A363345 A363345list[nmax_]:=With[{ca=CellularAutomaton[86,{{1},0},{{2nmax,3nmax-1},{nmax+1,2nmax}}]},Array[Length[FindRepeat[Diagonal[ca,nmax-#]]]&,nmax]];A363345list[200] (* Modified by _Paolo Xausa_, Aug 15 2023 *)
%o A363345 (Python) # See Brunnbauer link, Appendix 3.
%Y A363345 Cf. A070950, A094605 (periods of diagonals from the right).
%Y A363345 Cf. A363344 (diagonals), A363346 (lengths of transients), A364239 (indices of records), A364240.
%K A363345 nonn
%O A363345 1,4
%A A363345 _Paolo Xausa_, May 28 2023