%I #23 Jun 13 2022 21:20:13
%S 1,5,4,119,16,2015,64,32639,256,523775,1024,8386559,4096,134209535,
%T 16384,2147450879,65536,34359607295,262144,549755289599,1048576,
%U 8796090925055,4194304,140737479966719,16777216,2251799780130815,67108864,36028796884746239,268435456
%N Decimal representation of the n-th iteration of the "Rule 51" elementary cellular automaton starting with a single ON (black) cell.
%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H Robert Price, <a href="/A266668/b266668.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 03 2016 and Apr 18 2019: (Start)
%F a(n) = (-1/2-(-4)^n+(-2)^n+(-1)^n/2+4^n).
%F G.f.: (1+7*x-3*x^2+8*x^3+32*x^4) / ((1-x)*(1+x)*(1+2*x)*(1-4*x)*(1+4*x)).
%F (End)
%F Conjecture: a(n) = 2*4^n - 2^n - 1 for odd n; a(n) = 2^n for even n. - _Karl V. Keller, Jr._, Oct 12 2021
%t rule=51; 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. A266666, A266667.
%K nonn
%O 0,2
%A _Robert Price_, Jan 02 2016