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Decimal representation of the n-th iteration of the "Rule 203" elementary cellular automaton starting with a single ON (black) cell.
3

%I #37 Jul 06 2023 13:33:57

%S 1,4,27,119,495,2015,8127,32639,130815,523775,2096127,8386559,

%T 33550335,134209535,536854527,2147450879,8589869055,34359607295,

%U 137438691327,549755289599,2199022206975,8796090925055,35184367894527,140737479966719,562949936644095

%N Decimal representation of the n-th iteration of the "Rule 203" 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="/A267685/b267685.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 and Apr 17 2019: (Start)

%F a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>4.

%F G.f.: (1-3*x+13*x^2-22*x^3+8*x^4) / ((1-x)*(1-2*x)*(1-4*x)).

%F (End)

%F a(n) = A129868(n) for n >= 2. - _Georg Fischer_, Mar 26 2019

%F a(n) = 2*4^n - 2^n - 1 for n > 1. - _Karl V. Keller, Jr._, Jun 07 2022

%t 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 (Python) print([1, 4]+[2*4**n - 2**n - 1 for n in range(2, 50)]) # _Karl V. Keller, Jr._, Jun 07 2022

%Y Cf. A129868, A267683, A267684.

%K nonn,easy

%O 0,2

%A _Robert Price_, Jan 19 2016