A267705 Binary representation of the n-th iteration of the "Rule 205" elementary cellular automaton starting with a single ON (black) cell.

1, 10, 10101, 1101011, 111010111, 11110101111, 1111101011111, 111111010111111, 11111110101111111, 1111111101011111111, 111111111010111111111, 11111111110101111111111, 1111111111101011111111111, 111111111111010111111111111, 11111111111110101111111111111

Offset: 0

Author

Robert Price, Jan 19 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. A267704, A188530.

Programs

Mathematica

rule=205; 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) = 111*a(n-1)-1110*a(n-2)+1000*a(n-3) for n>3.

G.f.: (1-101*x+10101*x^2-10100*x^3) / ((1-x)*(1-10*x)*(1-100*x)).

(End)