A266069 Decimal representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.

1, 4, 2, 121, 4, 2035, 8, 32743, 16, 524239, 32, 8388511, 64, 134217535, 128, 2147483263, 256, 34359737599, 512, 549755812351, 1024, 8796093019135, 2048, 140737488349183, 4096, 2251799813672959, 8192, 36028797018939391, 16384, 576460752303374335, 32768

Offset

0,2

Comments

Rule 35 also generates this sequence.

References

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Robert Price, Table of n, a(n) for n = 0..999

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Formula

G.f.: (1+4*x-17*x^2+45*x^3+16*x^4-64*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)). - Colin Barker, Dec 21 2015 and Apr 18 2019

a(n) = 2*4^n - 3*2^((n-1)/2) - 1 for odd n; a(n) = 2^(n/2) for even n. - Karl V. Keller, Jr., Aug 25 2021

Example

From Michael De Vlieger, Dec 21 2015: (Start)
First 8 rows, replacing leading zeros with ".", the row converted to its binary, then decimal equivalent at right:
              1                =               1 ->     1
            1 0 0              =             100 ->     4
          . . . 1 0            =              10 ->     2
        1 1 1 1 0 0 1          =         1111001 ->   121
      . . . . . . 1 0 0        =             100 ->     4
    1 1 1 1 1 1 1 0 0 1 1      =     11111110011 ->  2035
  . . . . . . . . . 1 0 0 0    =            1000 ->     8
1 1 1 1 1 1 1 1 1 1 0 0 1 1 1  = 111111111100111 -> 32743
(End)

Mathematica

rule = 3; rows = 30; Table[FromDigits[Table[Take[CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}][[k]], {rows-k+1, rows+k-1}], {k, 1, rows}][[k]], 2], {k, 1, rows}]

Prog

(Python) print([2*4**n - 3*2**((n-1)//2) - 1 if n%2 else 2**(n//2) for n in range(30)]) # Karl V. Keller, Jr., Aug 25 2021

Crossrefs

Cf. A263428, A266068, A266070, A266071, A081253, A266072, A247375, A266073, A266074.

Sequence in context: A335300 A201228 A010319 * A057167 A355646 A096683

Adjacent sequences: A266066 A266067 A266068 * A266070 A266071 A266072

Keyword

nonn,easy

Author

Robert Price, Dec 20 2015

Status

approved