A267538 Binary representation of the middle column of the "Rule 143" elementary cellular automaton starting with a single ON (black) cell.
1, 11, 110, 1100, 11001, 110011, 1100111, 11001111, 110011111, 1100111111, 11001111111, 110011111111, 1100111111111, 11001111111111, 110011111111111, 1100111111111111, 11001111111111111, 110011111111111111, 1100111111111111111, 11001111111111111111
Offset: 0
Links
- Robert Price, Table of n, a(n) for n = 0..1000
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
- Index entries for sequences related to cellular automata
- Index to Elementary Cellular Automata
Programs
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Mathematica
rule=143; 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 *)
Formula
Conjectures from Colin Barker, Jan 17 2016 and Apr 20 2019: (Start)
a(n) = 11*a(n-1)-10*a(n-2) for n > 4. [n range correction by Karl V. Keller, Jr., Apr 23 2022]
G.f.: (1-x^2+x^4) / ((1-x)*(1-10*x)).
(End)
Conjecture: a(n) = floor(9901*10^n/9000). - Karl V. Keller, Jr., Apr 24 2022