%I #25 Feb 16 2025 08:33:29

%S 1,10,100,1000,10001,100010,1000101,10001010,100010101,1000101010,

%T 10001010101,100010101010,1000101010101,10001010101010,

%U 100010101010101,1000101010101010,10001010101010101,100010101010101010,1000101010101010101,10001010101010101010

%N Binary representation of the middle column of the "Rule 91" 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="/A267044/b267044.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://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 10 2016: (Start)

%F a(n) = (-550+450*(-1)^n+9901*10^n)/9900 for n>1.

%F a(n) = 10*a(n-1)+a(n-2)-10*a(n-3) for n>4. [Typo corrected by _Karl V. Keller, Jr._, Mar 16 2022]

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

%F (End)

%t rule=91; 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. A267015, A267045.

%K nonn

%O 0,2

%A _Robert Price_, Jan 09 2016

%E Removed an unjustified claim that _Colin Barker_'s conjectures are correct. Removed a program based on a conjecture. - _Michael De Vlieger_, Jun 13 2022