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Binary representation of the n-th iteration of the "Rule 103" elementary cellular automaton starting with a single ON (black) cell.
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%I #33 Jul 06 2023 13:13:31

%S 1,110,110,1110101,111100,11110001011,101111000,111111100010111,

%T 1011110000,1111111111000101111,10111100000,11111111111110001011111,

%U 101111000000,111111111111111100010111111,1011110000000,1111111111111111111000101111111,10111100000000

%N Binary representation of the n-th iteration of the "Rule 103" elementary cellular automaton starting with a single ON (black) cell.

%H Robert Price, <a href="/A267138/b267138.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 11 2016 and Apr 19 2019: (Start)

%F a(n) = 10011*a(n-2) - 110010*a(n-4) + 100000*a(n-6) for n > 10.

%F G.f.: (1 + 110*x - 9901*x^2 + 8891*x^3 - 880100*x^4 + 8881000*x^5 - 999110000*x^6 + 10991100000*x^7 - 999000000000*x^8 - 11000000000*x^9 + 1000000000000*x^10) / ((1-x)*(1+x)*(1-100*x)*(1+100*x)*(1-10*x^2)). (End)

%t rule=103; 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. A267136, A267139.

%K nonn,easy

%O 0,2

%A _Robert Price_, Jan 10 2016

%E Removed an unjustified claim that _Colin Barker_'s conjectures are correct. Removed a program based on a conjecture. - _Michael De Vlieger_, Jun 13 2022