A267614 Decimal representation of the n-th iteration of the "Rule 185" elementary cellular automaton starting with a single ON (black) cell.

1, 1, 11, 47, 191, 767, 3071, 12287, 49151, 196607, 786431, 3145727, 12582911, 50331647, 201326591, 805306367, 3221225471, 12884901887, 51539607551, 206158430207, 824633720831, 3298534883327, 13194139533311, 52776558133247, 211106232532991, 844424930131967

Offset: 0

Author

Robert Price, Jan 18 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. A198693, A267612, A267613.

Programs

Mathematica

rule=185; 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]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Formula

Conjectures from Colin Barker, Jan 18 2016 and Apr 20 2019: (Start)

a(n) = 5*a(n-1) - 4*a(n-2) for n>3.

G.f.: (1-4*x+10*x^2-4*x^3) / ((1-x)*(1-4*x)).

(End)

Empirical a(n) = 3*4^(n-1)-1 = A198693(n-1) for n>1. - Colin Barker, Nov 25 2016 and Apr 20 2019

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