A267811 Binary representation of the n-th iteration of the "Rule 217" elementary cellular automaton starting with a single ON (black) cell.

1, 1, 11011, 1110111, 111101111, 11111011111, 1111110111111, 111111101111111, 11111111011111111, 1111111110111111111, 111111111101111111111, 11111111111011111111111, 1111111111110111111111111, 111111111111101111111111111, 11111111111111011111111111111

Offset: 0

Author

Robert Price, Jan 20 2016

Keywords

• nonn
• easy

Comments

Conjectures from Barker confirmed by later formula. - Ray Chandler, Aug 09 2025

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
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. A267810.

Cf. A138148, A267684.

Programs

Mathematica

rule=217; 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

From Colin Barker, Jan 22 2016 and Apr 20 2019: (Start)

a(n) = 111*a(n-1)-1110*a(n-2)+1000*a(n-3) for n>4.

G.f.: (1-110*x+12010*x^2-112000*x^3+100000*x^4) / ((1-x)*(1-10*x)*(1-100*x)).

(End)

a(n) = A138148(n) for n > 1. - Ray Chandler, Aug 09 2025