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Decimal representation of the n-th iteration of the "Rule 201" elementary cellular automaton starting with a single ON (black) cell.
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%I #29 Jun 14 2022 01:23:11

%S 1,0,21,99,471,1935,8031,32319,130431,522495,2094591,8381439,33544191,

%T 134189055,536829951,2147368959,8589770751,34359279615,137438298111,

%U 549753978879,2199020634111,8796085682175,35184361603071,140737458995199,562949911478271

%N Decimal representation of the n-th iteration of the "Rule 201" elementary cellular automaton starting with a single ON (black) cell.

%C The fact that only n cells to the left and right of the initially active cell are updated at step n (cf. A267679 for the binary representation) is contradictory to the usual treatment of a "Rule m" automaton, where all cells are updated depending on their neighborhood. See also the illustration of "Rule 201" on the MathWorld page. - _M. F. Hasler_, Jul 28 2018

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

%H Robert Price, <a href="/A267681/b267681.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 19 2016: (Start)

%F a(n) = 5*a(n-1)-20*a(n-3)+16*a(n-4) for n>4.

%F G.f.: (1-5*x+21*x^2+14*x^3-40*x^4) / ((1-x)*(1-2*x)*(1+2*x)*(1-4*x)).

%F (End)

%F Conjecture: a(n) = 2*4^n - (n%2*2 + [n]*5)*2^(n-1) - 1, where [n] = 1 iff n > 0; n%2 = 1 iff n is odd. - _M. F. Hasler_, Jul 28 2018

%t rule=201; 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. A267679, A267680.

%K nonn

%O 0,3

%A _Robert Price_, Jan 19 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