A266299 Binary representation of the n-th iteration of the "Rule 14" elementary cellular automaton starting with a single ON (black) cell.

1, 110, 11000, 1100000, 110000000, 11000000000, 1100000000000, 110000000000000, 11000000000000000, 1100000000000000000, 110000000000000000000, 11000000000000000000000, 1100000000000000000000000, 110000000000000000000000000, 11000000000000000000000000000

Offset: 0

Author

Robert Price, Dec 26 2015

Keywords

• nonn
• easy

Comments

Rules 46, 142 and 174 also generate this sequence.

Links

• Robert Price, Table of n, a(n) for n = 0..999
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata
• Index entries for linear recurrences with constant coefficients, signature (100).

Crossrefs

Cf. A164908, A266298.

Programs

Mathematica

rule=14; 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 *)
LinearRecurrence[{100},{1,110},20] (* Harvey P. Dale, Aug 27 2023 *)

Python

print([int(110*100**(n-1)) for n in range(50)]) # Karl V. Keller, Jr., Aug 30 2021

Formula

From Colin Barker, Dec 28 2015 and Apr 14 2019: (Start)

a(n) = 11*10^(2*n-1) for n>0.

a(n) = 100*a(n-1) for n>1.

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

(End)

a(n) = floor(110*100^(n-1)). - Karl V. Keller, Jr., Aug 30 2021