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Decimal representation of the n-th iteration of the "Rule 141" elementary cellular automaton starting with a single ON (black) cell.
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%I #23 Sep 25 2021 15:06:41

%S 1,2,5,11,87,175,1375,2751,21887,43775,349695,699391,5593087,11186175,

%T 89481215,178962431,1431666687,2863333375,22906535935,45813071871,

%U 366504050687,733008101375,5864062713855,11728125427711,93824995033087,187649990066175

%N Decimal representation of the n-th iteration of the "Rule 141" 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="/A267527/b267527.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 From _Colin Barker_, Jan 16 2016 and Apr 19 2019: (Start)

%F a(n) = 3*a(n-1)+14*a(n-2)-48*a(n-3)+32*a(n-4) for n>5.

%F G.f.: (1-x-15*x^2+16*x^3+48*x^4-64*x^5) / ((1-x)*(1-2*x)*(1-4*x)*(1+4*x)).

%F (End)

%F a(n) = 2^n*(4^floor(n/2) - 1)/3 + 2^(n-1) - 1 for n > 1. - _Karl V. Keller, Jr._, Sep 22 2021

%t rule=141; 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, 2] + [2**n*(4**(n//2) - 1)//3 + 2**(n-1) - 1 for n in range(2,50)]) # _Karl V. Keller, Jr._, Sep 22 2021

%Y Cf. A267525, A267526.

%K nonn,easy

%O 0,2

%A _Robert Price_, Jan 16 2016