A267777 Binary representation of the n-th iteration of the "Rule 209" elementary cellular automaton starting with a single ON (black) cell.

1, 1, 11001, 1111001, 111111001, 11111111001, 1111111111001, 111111111111001, 11111111111111001, 1111111111111111001, 111111111111111111001, 11111111111111111111001, 1111111111111111111111001, 111111111111111111111111001, 11111111111111111111111111001

Offset: 0

Author

Robert Price, Jan 20 2016

Keywords

• nonn
• easy

Comments

Rule 241 also generates this sequence.

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. A267776, A141722.

Programs

Mathematica

rule=209; 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 *) Table[FromDigits[catri[[k]]],{k,1,rows}]   (* Binary Representation of Rows *)

Formula

Conjectures from Colin Barker, Jan 20 2016 and Apr 20 2019: (Start)

a(n) = 101*a(n-1)-100*a(n-2) for n>2.

G.f.: (1-100*x+11000*x^2) / ((1-x)*(1-100*x)).

(End)

Extensions

Removed an unjustified claim that Colin Barker's conjectures are correct. Removed a program based on a conjecture. - Michael De Vlieger, Jun 13 2022