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Binary representation of the n-th iteration of the "Rule 67" elementary cellular automaton starting with a single ON (black) cell.
3

%I #24 Jul 06 2023 13:03:18

%S 1,100,1010,1100001,1011100,11100001011,1011100000,111100001011111,

%T 1011100000000,1111100001011111111,1011100000000000,

%U 11111100001011111111111,1011100000000000000,111111100001011111111111111,1011100000000000000000,1111111100001011111111111111111

%N Binary representation of the n-th iteration of the "Rule 67" elementary cellular automaton starting with a single ON (black) cell.

%H Robert Price, <a href="/A266838/b266838.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://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) = 11001*a(n-2) - 10011000*a(n-4) + 10000000*a(n-6) for n > 9.

%F G.f.: (1 + 100*x - 9991*x^2 - 99*x^3 - 88910*x^4 - 9990*x^5 - 10901100*x^6 - 99900*x^7 + 11000000*x^8 - 1000000*x^9) / ((1-x)*(1+x)*(1-100*x)*(1+100*x)*(1-1000*x^2)).

%F (End)

%F Conjecture: a(n) = (10*100^n - 999909*1000^floor(n/2)/10000 - 1)/9 for odd n > 3; a(n) = 10111*1000^(n/2)/10000 for even n > 3. - _Karl V. Keller, Jr._, Dec 16 2021

%t rule=67; 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. A266837, A266839.

%K nonn,easy

%O 0,2

%A _Robert Price_, Jan 04 2016