A266612 Binary representation of the middle column of the "Rule 41" elementary cellular automaton starting with a single ON (black) cell.
1, 10, 101, 1010, 10100, 101001, 1010010, 10100101, 101001010, 1010010101, 10100101010, 101001010101, 1010010101010, 10100101010101, 101001010101010, 1010010101010101, 10100101010101010, 101001010101010101, 1010010101010101010, 10100101010101010101
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=41; 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 02 2016 and Apr 17 2019: (Start)
a(n) = (-9*(-1)^n+99991*2^(n-2)*5^(n-3)-11)/198 for n>2.
a(n) = 10*a(n-1)+a(n-2)-10*a(n-3) for n>2.
G.f.: (1-x^4+x^5) / ((1-x)*(1+x)*(1-10*x)).
(End)
Extensions
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