A265723 Number of OFF (white) cells in the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

0, 3, 4, 3, 8, 3, 12, 3, 16, 3, 20, 3, 24, 3, 28, 3, 32, 3, 36, 3, 40, 3, 44, 3, 48, 3, 52, 3, 56, 3, 60, 3, 64, 3, 68, 3, 72, 3, 76, 3, 80, 3, 84, 3, 88, 3, 92, 3, 96, 3, 100, 3, 104, 3, 108, 3, 112, 3, 116, 3, 120, 3, 124, 3, 128, 3, 132, 3, 136, 3, 140, 3

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..999

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Formula

Conjectures from Colin Barker, Dec 15 2015 and Apr 16 2019: (Start)

a(n) = 1/2*((2*(-1)^n+2)*n-3*((-1)^n-1)).

a(n) = 2*a(n-2) - a(n-4) for n>3.

G.f.: x*(3+4*x-3*x^2) / ((1-x)^2*(1+x)^2).

(End)

Example

From Michael De Vlieger, Dec 14 2015: (Start)
First 12 rows, replacing ones with "." for better visibility of OFF cells, followed by the total number of 0's per row at right:
                      .                        =  0
                    0 0 0                      =  3
                  0 0 . 0 0                    =  4
                . . 0 0 0 . .                  =  3
              0 0 0 0 . 0 0 0 0                =  8
            . . . . 0 0 0 . . . .              =  3
          0 0 0 0 0 0 . 0 0 0 0 0 0            = 12
        . . . . . . 0 0 0 . . . . . .          =  3
      0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0        = 16
    . . . . . . . . 0 0 0 . . . . . . . .      =  3
  0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0    = 20
. . . . . . . . . . 0 0 0 . . . . . . . . . .  =  3
(End)

Mathematica

rows = 71; Count[#, n_ /; n == 0] & /@ Table[Table[Take[CellularAutomaton[1, {{1}, 0}, rows - 1, {All, All}][[k]], {rows - k + 1, rows + k - 1}], {k, 1, rows}][[k]], {k, 1, rows}] (* Michael De Vlieger, Dec 14 2015 *)

Crossrefs

Cf. A265718, A265720, A265721.

Sequence in context: A350771 A077005 A328347 * A134065 A015887 A158134

Adjacent sequences: A265720 A265721 A265722 * A265724 A265725 A265726

Keyword

nonn,easy

Author

Robert Price, Dec 14 2015

Status

approved