A267524 Binary representation of the middle column of the "Rule 139" elementary cellular automaton starting with a single ON (black) cell.

1, 10, 100, 1001, 10011, 100111, 1001111, 10011111, 100111111, 1001111111, 10011111111, 100111111111, 1001111111111, 10011111111111, 100111111111111, 1001111111111111, 10011111111111111, 100111111111111111, 1001111111111111111, 10011111111111111111

Offset: 0

Author

Robert Price, Jan 16 2016

Keywords

• nonn
• easy

References

• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

• Robert Price, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• S. Wolfram, A New Kind of Science
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A267520, A054135.

Programs

Mathematica

rule=139; 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 16 2016 and Apr 19 2019: (Start)

a(n) = 11*a(n-1) - 10*a(n-2) for n > 3. [n range correction - Karl V. Keller, Jr., Apr 08 2022]

G.f.: (1-x+x^3) / ((1-x)*(1-10*x)).

(End)