A266844 Binary representation of the n-th iteration of the "Rule 70" elementary cellular automaton starting with a single ON (black) cell.

1, 110, 10100, 1101000, 101010000, 11010100000, 1010101000000, 110101010000000, 10101010100000000, 1101010101000000000, 101010101010000000000, 11010101010100000000000, 1010101010101000000000000, 110101010101010000000000000, 10101010101010100000000000000

Offset: 0

Author

Robert Price, Jan 04 2016

Keywords

• nonn
• easy

Comments

Rule 198 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. A266843, A266846.

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]]],{k,1,rows}]   (* Binary Representation of Rows *)

Formula

Conjectures from Colin Barker, Jan 05 2016 and Apr 18 2019: (Start)

a(n) = (2^(n-1)*(-9*(-50)^n-2*5^n+209*50^n))/99.

a(n) = 10*a(n-1)+10000*a(n-2)-100000*a(n-3) for n>2.

G.f.: (1+100*x-1000*x^2) / ((1-10*x)*(1-100*x)*(1+100*x)).

(End)