A266219 Binary representation of the middle column of the "Rule 7" elementary cellular automaton starting with a single ON (black) cell.

1, 11, 110, 1101, 11010, 110101, 1101010, 11010101, 110101010, 1101010101, 11010101010, 110101010101, 1101010101010, 11010101010101, 110101010101010, 1101010101010101, 11010101010101010, 110101010101010101, 1101010101010101010, 11010101010101010101

Offset: 0

Author

Robert Price, Dec 24 2015

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..499
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A266216, A081254.

Programs

Mathematica

rule=7; 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 *) mc=Table[catri[[k]][[k]],{k,1,rows}]; (* Keep only middle cell from each row *) Table[FromDigits[Take[mc,k]],{k,1,rows}]  (* Binary Representation of Middle Column *)

Formula

Conjectures from Colin Barker, Dec 25 2015 and Apr 13 2019: (Start)

a(n) = 1/198*(-9*(-1)^n+109*2^(n+1)*5^n-11).

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

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

(End)

Conjecture: a(n) = floor(109*10^n/99). - Karl V. Keller, Jr., Mar 16 2022