A266667 Binary representation of the n-th iteration of the "Rule 51" elementary cellular automaton starting with a single ON (black) cell.

1, 101, 100, 1110111, 10000, 11111011111, 1000000, 111111101111111, 100000000, 1111111110111111111, 10000000000, 11111111111011111111111, 1000000000000, 111111111111101111111111111, 100000000000000, 1111111111111110111111111111111, 10000000000000000

Offset

0,2

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

Formula

Conjectures from Colin Barker, Jan 03 2016 and Apr 18 2019: (Start)

a(n) = (-1+(-1)^n+9*(-5)^n*2^(1+n)+10^(1+2*n)-(-1)^n*10^(1+2*n))/18.

G.f.: (1+111*x-8891*x^2+1000*x^3+20000*x^4) / ((1-x)*(1+x)*(1+10*x)*(1-100*x)*(1+100*x)).

(End)

Conjecture: a(n) = (10*100^n - 9*10^n - 1)/9 for odd n; a(n) = 10^n for even n. - Karl V. Keller, Jr., Oct 12 2021

Mathematica

rule=51; 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 *)

Crossrefs

Cf. A266666, A266668.

Sequence in context: A281841 A282223 A282200 * A368397 A241495 A282980

Adjacent sequences: A266664 A266665 A266666 * A266668 A266669 A266670

Keyword

nonn

Author

Robert Price, Jan 02 2016

Status

approved