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Decimal representation of the n-th iteration of the "Rule 123" elementary cellular automaton starting with a single ON (black) cell.
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%I #29 Feb 16 2025 08:33:29

%S 1,5,14,119,56,2015,224,32639,896,523775,3584,8386559,14336,134209535,

%T 57344,2147450879,229376,34359607295,917504,549755289599,3670016,

%U 8796090925055,14680064,140737479966719,58720256,2251799780130815,234881024,36028796884746239

%N Decimal representation of the n-th iteration of the "Rule 123" elementary cellular automaton starting with a single ON (black) cell.

%H Robert Price, <a href="/A267351/b267351.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 14 2016 and Apr 19 2019: (Start)

%F a(n) = 21*a(n-2)-84*a(n-4)+64*a(n-6) for n>6.

%F G.f.: (1+5*x-7*x^2+14*x^3-154*x^4-64*x^5+160*x^6) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-4*x)*(1+4*x)).

%F (End)

%t rule=123; 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]],2],{k,1,rows}] (* Decimal Representation of Rows *)

%Y Cf. A267349, A267350.

%K nonn,easy,changed

%O 0,2

%A _Robert Price_, Jan 13 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