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Decimal representation of the n-th iteration of the "Rule 85" elementary cellular automaton starting with a single ON (black) cell.
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%I #27 Jul 06 2023 13:12:43

%S 1,3,16,63,256,1023,4096,16383,65536,262143,1048576,4194303,16777216,

%T 67108863,268435456,1073741823,4294967296,17179869183,68719476736,

%U 274877906943,1099511627776,4398046511103,17592186044416,70368744177663,281474976710656

%N Decimal representation of the n-th iteration of the "Rule 85" elementary cellular automaton starting with a single ON (black) cell.

%H Robert Price, <a href="/A267036/b267036.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) = ((-1)^n-1)/2+4^n.

%F a(n) = 4*a(n-1)+a(n-2)-4*a(n-3) for n>2.

%F G.f.: (1-x+3*x^2) / ((1-x)*(1+x)*(1-4*x)).

%F (End)

%t rule=85; 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. A267034, A267035.

%K nonn,easy

%O 0,2

%A _Robert Price_, Jan 09 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