A267514 - OEIS

%I #14 Apr 12 2019 06:05:07

%S 1,10,101,1010,10101,101010,1010100,10101001,101010010,1010100101,

%T 10101001010,101010010101,1010100101010,10101001010100,

%U 101010010101001,1010100101010010,10101001010100101,101010010101001010,1010100101010010101,10101001010100101010

%N Binary representation of the middle column of the "Rule 137" 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="/A267514/b267514.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 a(n) = A007088(A267515(n)). - _R. J. Mathar_, Apr 12 2019

%F Conjectures from _Colin Barker_, Jan 16 2016: (Start)

%F a(n) = 10*a(n-1)+a(n-7)-10*a(n-8) for n>7.

%F G.f.: (1-x+x^2)*(1+x+x^2) / ((1-x)*(1-10*x)*(1+x+x^2+x^3+x^4+x^5+x^6)).

%F (End)

%F This conjectured linear recurrence fails to predict the correct values from a(62) on. - _R. J. Mathar_, Apr 12 2019

%t rule=137; 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 *) mc=Table[catri[[k]][[k]],{k,1,rows}]; (* Keep only middle cell from each row *) Table[FromDigits[Take[mc,k]],{k,1,rows}]  (* Binary Representation of Middle Column *)

%Y Cf. A267463.

%K nonn,easy

%O 0,2

%A _Robert Price_, Jan 16 2016