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A118109 Binary representation of n-th iteration of the Rule 54 elementary cellular automaton starting with a single black cell.

%I A118109 #53 Feb 16 2025 08:33:00
%S A118109 1,111,10001,1110111,100010001,11101110111,1000100010001,
%T A118109 111011101110111,10001000100010001,1110111011101110111,
%U A118109 100010001000100010001,11101110111011101110111,1000100010001000100010001,111011101110111011101110111,10001000100010001000100010001,1110111011101110111011101110111
%N A118109 Binary representation of n-th iteration of the Rule 54 elementary cellular automaton starting with a single black cell.
%H A118109 Robert Price, <a href="/A118109/b118109.txt">Table of n, a(n) for n = 0..499</a>
%H A118109 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule54.html">Rule 54</a>
%H A118109 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A118109 Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>, Wolfram Media, 2002; p. 55.
%H A118109 <a href="https://oeis.org/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A118109 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A118109 Conjectures from _Colin Barker_, Dec 08 2015 and Apr 16 2019: (Start)
%F A118109 a(n) = 10001*a(n-2)-10000*a(n-4) for n>3.
%F A118109 G.f.: (1+111*x) / ((1-x)*(1+x)*(1-100*x)*(1+100*x)).
%F A118109 (End)
%F A118109 Conjecture: a(n) = floor((10000+1100*(n mod 2))*100^n/9999). - _Karl V. Keller, Jr._, Sep 24 2021
%e A118109 From _Michael De Vlieger_, Oct 07 2015: (Start)
%e A118109 First 8 rows, representing ON cells as "1", OFF cells within the bounds of ON cells as "0", interpreted as a binary number at left, the decimal equivalent appearing at right (A118108):
%e A118109                    1 =     1
%e A118109                  111 =     7
%e A118109               1 0001 =    17
%e A118109             111 0111 =   119
%e A118109          1 0001 0001 =   273
%e A118109        111 0111 0111 =  1911
%e A118109     1 0001 0001 0001 =  4369
%e A118109   111 0111 0111 0111 = 30583
%e A118109 10001 0001 0001 0001 = 69905
%e A118109 (End)
%t A118109 rule=54; 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 *) (* _Robert Price_, Feb 21 2016 *)
%Y A118109 Cf. A071030 (essentially the same but lists bits separately), A118108 (converted to base 10).
%K A118109 nonn,easy
%O A118109 0,2
%A A118109 _Eric W. Weisstein_, Apr 13 2006
%E A118109 Terms changed to match definition, as suggested by _Michael De Vlieger_. - _N. J. A. Sloane_, Oct 17 2015