A265720 Binary representation of the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

1, 0, 100, 1100011, 10000, 11110001111, 1000000, 111111000111111, 100000000, 1111111100011111111, 10000000000, 11111111110001111111111, 1000000000000, 111111111111000111111111111, 100000000000000, 1111111111111100011111111111111, 10000000000000000

Offset

0,3

Comments

Rule 33 also generates 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 (0,10101,0,-1010100,0,1000000).

Formula

From Colin Barker, Dec 14 2015 and Apr 16 2019: (Start)

a(n) = 10101*a(n-2) - 1010100*a(n-4) + 1000000*a(n-6) for n > 5.

G.f.: (1-10001*x^2+1100011*x^3+10000*x^4-1210000*x^5) / ((1-x)*(1+x)*(1-10*x)*(1+10*x)*(1-100*x)*(1+100*x)).

(End)

a(n) = (10*100^n - 999*10^(n-1) - 1)/9 for odd n; a(n) = 10^n for even n. - Karl V. Keller, Jr., Aug 25 2021

Example

From Michael De Vlieger, Dec 14 2015: (Start)
First 10 rows, replacing leading zeros with ".", the row converted to its binary equivalent at right:
                  1                    =                   1
                . . 0                  =                   0
              . . 1 0 0                =                 100
            1 1 0 0 0 1 1              =             1100011
          . . . . 1 0 0 0 0            =               10000
        1 1 1 1 0 0 0 1 1 1 1          =         11110001111
      . . . . . . 1 0 0 0 0 0 0        =             1000000
    1 1 1 1 1 1 0 0 0 1 1 1 1 1 1      =     111111000111111
  . . . . . . . . 1 0 0 0 0 0 0 0 0    =           100000000
1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1  = 1111111100011111111
(End)

Mathematica

rule = 1; rows = 20; Table[FromDigits[Table[Take[CellularAutomaton[rule, {{1}, 0}, rows - 1, {All, All}][[k]], {rows - k + 1, rows + k - 1}], {k, 1, rows}][[k]]], {k, 1, rows}]

Prog

(Python) print([(10*100**n - 999*10**(n-1) - 1)//9 if n%2 else 10**n for n in range(50)]) # Karl V. Keller, Jr., Aug 25 2021

Crossrefs

Cf. A265718, A265721.

Sequence in context: A202055 A185303 A139110 * A343182 A013855 A204576

Adjacent sequences: A265717 A265718 A265719 * A265721 A265722 A265723

Keyword

nonn,easy

Author

Robert Price, Dec 14 2015

Status

approved