A266846 Decimal representation of the n-th iteration of the "Rule 70" elementary cellular automaton starting with a single ON (black) cell.

1, 6, 20, 104, 336, 1696, 5440, 27264, 87296, 436736, 1397760, 6989824, 22368256, 111845376, 357908480, 1789558784, 5726601216, 28633071616, 91625881600, 458129670144, 1466015154176, 7330076819456, 23456246661120, 117281237499904, 375299963355136

Offset: 0

Author

Robert Price, Jan 04 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. A266843, A266844.

Programs

Mathematica

rule=70; 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 05 2016 and Apr 18 2019: (Start)

a(n) = 2^(n-1)*(-(-2)^n+9*2^n-2)/3 = 2^(n-1)*A061547(n+3).

a(n) = 2*a(n-1)+16*a(n-2)-32*a(n-3) for n>2.

G.f.: (1+4*x-8*x^2) / ((1-2*x)*(1-4*x)*(1+4*x)).

(End)