A266508 Binary representation of the n-th iteration of the "Rule 28" elementary cellular automaton starting with a single ON (black) cell.

1, 11, 101, 1011, 10101, 101011, 1010101, 10101011, 101010101, 1010101011, 10101010101, 101010101011, 1010101010101, 10101010101011, 101010101010101, 1010101010101011, 10101010101010101, 101010101010101011, 1010101010101010101, 10101010101010101011

Offset: 0

Author

Robert Price, Dec 30 2015

Keywords

• nonn
• easy

Comments

Rule 156 also generates this sequence.

Also, the binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 678", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283641.

Links

• Robert Price, Table of n, a(n) for n = 0..1000
• 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 (10,1,-10).

Crossrefs

Cf. A266502, A283641, A283642.

Programs

Mathematica

rule=28; 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 *)

Python

print([(100*10**n + 89)//99 for n in range(50)]) # Karl V. Keller, Jr., Sep 04 2021

Formula

From Colin Barker, Dec 30 2015 and Apr 16 2019: (Start)

a(n) = (44 - 45*(-1)^n + 10^(2+n))/99.

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

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

(End)

a(n) = floor((100*10^n + 89)/99). - Karl V. Keller, Jr., Sep 04 2021