A265722 Number of ON (black) cells in the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

1, 0, 1, 4, 1, 8, 1, 12, 1, 16, 1, 20, 1, 24, 1, 28, 1, 32, 1, 36, 1, 40, 1, 44, 1, 48, 1, 52, 1, 56, 1, 60, 1, 64, 1, 68, 1, 72, 1, 76, 1, 80, 1, 84, 1, 88, 1, 92, 1, 96, 1, 100, 1, 104, 1, 108, 1, 112, 1, 116, 1, 120, 1, 124, 1, 128, 1, 132, 1, 136, 1, 140

Offset

0,4

References

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

Links

Colin Barker, Table of n, a(n) for n = 0..1000

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 14 2015 and Apr 16 2019: (Start)

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

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

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

(End)

a(n) = A019425(n), n>1. - R. J. Mathar, Jan 10 2016

Example

From Michael De Vlieger, Dec 14 2015: (Start)
First 12 rows, replacing zeros with "." for better visibility of ON cells, followed by the total number of 1's per row at right:
                      1                        =   1
                    . . .                      =   0
                  . . 1 . .                    =   1
                1 1 . . . 1 1                  =   4
              . . . . 1 . . . .                =   1
            1 1 1 1 . . . 1 1 1 1              =   8
          . . . . . . 1 . . . . . .            =   1
        1 1 1 1 1 1 . . . 1 1 1 1 1 1          =  12
      . . . . . . . . 1 . . . . . . . .        =   1
    1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1      =  16
  . . . . . . . . . . 1 . . . . . . . . . .    =   1
1 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 1  =  20
(End)

Mathematica

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

Crossrefs

Cf. A265718, A265720, A265721.

Sequence in context: A211779 A318445 A158496 * A019425 A255242 A329371

Adjacent sequences: A265719 A265720 A265721 * A265723 A265724 A265725

Keyword

nonn,easy

Author

Robert Price, Dec 14 2015

Status

approved