%I #25 Feb 16 2025 08:33:28
%S 1,10,10100,101011,101010000,1010101111,1010101000000,10101010111111,
%T 10101010100000000,101010101011111111,101010101010000000000,
%U 1010101010101111111111,1010101010101000000000000,10101010101010111111111111,10101010101010100000000000000
%N Binary representation of the n-th iteration of the "Rule 69" elementary cellular automaton starting with a single ON (black) cell.
%H Robert Price, <a href="/A266841/b266841.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>, Wolfram Media, 2002; p. 55.
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F Conjectures from _Colin Barker_, Jan 05 2016 and Apr 18 2019: (Start)
%F a(n) = 10101*a(n-2)-1010100*a(n-4)+1000000*a(n-6) for n>5.
%F G.f.: (1+10*x-x^2+x^3-110000*x^5) / ((1-x)*(1+x)*(1-10*x)*(1+10*x)*(1-100*x)*(1+100*x)).
%F (End)
%F Conjecture: a(n) = ( 10*100^n + 100^floor(n/2) - 11)/99 for odd n; a(n) = (100*100^n - 10^n)/99 for even n. - _Karl V. Keller, Jr._, Feb 28 2022
%t rule=69; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]]],{k,1,rows}] (* Binary Representation of Rows *)
%Y Cf. A266840, A266842.
%K nonn,easy,changed
%O 0,2
%A _Robert Price_, Jan 04 2016