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Binary representation of the n-th iteration of the "Rule 51" elementary cellular automaton starting with a single ON (black) cell.
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%I #23 Jun 13 2022 21:19:24

%S 1,101,100,1110111,10000,11111011111,1000000,111111101111111,

%T 100000000,1111111110111111111,10000000000,11111111111011111111111,

%U 1000000000000,111111111111101111111111111,100000000000000,1111111111111110111111111111111,10000000000000000

%N Binary representation of the n-th iteration of the "Rule 51" elementary cellular automaton starting with a single ON (black) cell.

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

%H Robert Price, <a href="/A266667/b266667.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 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

%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 03 2016 and Apr 18 2019: (Start)

%F a(n) = (-1+(-1)^n+9*(-5)^n*2^(1+n)+10^(1+2*n)-(-1)^n*10^(1+2*n))/18.

%F G.f.: (1+111*x-8891*x^2+1000*x^3+20000*x^4) / ((1-x)*(1+x)*(1+10*x)*(1-100*x)*(1+100*x)).

%F (End)

%F Conjecture: a(n) = (10*100^n - 9*10^n - 1)/9 for odd n; a(n) = 10^n for even n. - _Karl V. Keller, Jr._, Oct 12 2021

%t rule=51; 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. A266666, A266668.

%K nonn

%O 0,2

%A _Robert Price_, Jan 02 2016