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A267623 Binary representation of the middle column of the "Rule 187" elementary cellular automaton starting with a single ON (black) cell.

%I A267623 #49 Feb 16 2025 08:33:29
%S A267623 1,10,101,1011,10111,101111,1011111,10111111,101111111,1011111111,
%T A267623 10111111111,101111111111,1011111111111,10111111111111,
%U A267623 101111111111111,1011111111111111,10111111111111111,101111111111111111,1011111111111111111,10111111111111111111
%N A267623 Binary representation of the middle column of the "Rule 187" elementary cellular automaton starting with a single ON (black) cell.
%C A267623 Also, The binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283508.
%D A267623 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A267623 Robert Price, <a href="/A267623/b267623.txt">Table of n, a(n) for n = 0..1000</a>
%H A267623 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A267623 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A267623 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A267623 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A267623 Conjectures from _Colin Barker_, Jan 19 2016 and Apr 16 2019: (Start)
%F A267623 a(n) = 11*a(n-1)-10*a(n-2) for n>2.
%F A267623 G.f.: (1-x+x^2) / ((1-x)*(1-10*x)).
%F A267623 (End)
%F A267623 Empirical: a(n) = (91*10^n - 10) / 90 for n>0. - _Colin Barker_, Mar 10 2017
%F A267623 It also appears that a(n) = floor(91*10^n/90). - _Karl V. Keller, Jr._, May 28 2022
%p A267623 # Rule 187: value in generation r and column c, where c=0 is the central one
%p A267623 r187 := proc(r::integer,c::integer)
%p A267623     option remember;
%p A267623     local up ;
%p A267623     if r = 0 then
%p A267623         if c = 0 then
%p A267623             1;
%p A267623         else
%p A267623             0;
%p A267623         end if;
%p A267623     else
%p A267623         # previous 3 bits
%p A267623         [procname(r-1,c+1),procname(r-1,c),procname(r-1,c-1)] ;
%p A267623         up := op(3,%)+2*op(2,%)+4*op(1,%) ;
%p A267623         # rule 187 = 10111011_2: {6,2}->0, all others ->1
%p A267623         if up in {6,2} then
%p A267623             0;
%p A267623         else
%p A267623             1 ;
%p A267623         end if;
%p A267623     end if;
%p A267623 end proc:
%p A267623 A267623 := proc(n)
%p A267623     b := [seq(r187(r,0),r=0..n)] ;
%p A267623     add(op(-i,b)*2^(i-1),i=1..nops(b)) ;
%p A267623     A007088(%) ;
%p A267623 end proc:
%p A267623 smax := 30 ;
%p A267623 L := [seq(A267623(n),n=0..smax)] ; # _R. J. Mathar_, Apr 12 2019
%t A267623 rule=187; 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 A267623 Cf. A267621, A283508, A083329.
%K A267623 nonn,easy
%O A267623 0,2
%A A267623 _Robert Price_, Jan 18 2016
%E A267623 Removed an unjustified claim that _Colin Barker_'s conjectures are correct. Removed a program based on a conjecture. - _N. J. A. Sloane_, Jun 13 2022