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A267685 Decimal representation of the n-th iteration of the "Rule 203" elementary cellular automaton starting with a single ON (black) cell.

%I A267685 #41 Aug 09 2025 10:00:03
%S A267685 1,4,27,119,495,2015,8127,32639,130815,523775,2096127,8386559,
%T A267685 33550335,134209535,536854527,2147450879,8589869055,34359607295,
%U A267685 137438691327,549755289599,2199022206975,8796090925055,35184367894527,140737479966719,562949936644095
%N A267685 Decimal representation of the n-th iteration of the "Rule 203" elementary cellular automaton starting with a single ON (black) cell.
%C A267685 Conjectures from Barker confirmed by later formulas. - _Ray Chandler_, Aug 09 2025
%D A267685 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A267685 Robert Price, <a href="/A267685/b267685.txt">Table of n, a(n) for n = 0..1000</a>
%H A267685 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A267685 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A267685 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A267685 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A267685 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).
%F A267685 From _Colin Barker_, Jan 19 2016 and Apr 17 2019: (Start)
%F A267685 a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>4.
%F A267685 G.f.: (1-3*x+13*x^2-22*x^3+8*x^4) / ((1-x)*(1-2*x)*(1-4*x)).
%F A267685 (End)
%F A267685 a(n) = A129868(n) for n >= 2. - _Georg Fischer_, Mar 26 2019
%F A267685 a(n) = 2*4^n - 2^n - 1 for n > 1. - _Karl V. Keller, Jr._, Jun 07 2022
%t A267685 rule=203; 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 *)
%o A267685 (Python) print([1, 4]+[2*4**n - 2**n - 1 for n in range(2, 50)]) # _Karl V. Keller, Jr._, Jun 07 2022
%Y A267685 Cf. A129868, A267683, A267684.
%K A267685 nonn,easy
%O A267685 0,2
%A A267685 _Robert Price_, Jan 19 2016