A265724 Total number of OFF (white) cells after n iterations of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

0, 3, 7, 10, 18, 21, 33, 36, 52, 55, 75, 78, 102, 105, 133, 136, 168, 171, 207, 210, 250, 253, 297, 300, 348, 351, 403, 406, 462, 465, 525, 528, 592, 595, 663, 666, 738, 741, 817, 820, 900, 903, 987, 990, 1078, 1081, 1173, 1176, 1272, 1275, 1375, 1378, 1482

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 16 2015 and Apr 16 2019: (Start)

a(n) = 1/2*(n^2+(-1)^n*n+4*n-(-1)^n+1).

a(n) = 1/2*(n^2+5*n) for n even.

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

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.

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

(End)

Apparently, a(n) = A267049(n) + 4*floor(n/2) - 1 for n>1. - Hugo Pfoertner, Jun 21 2024

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, and the running total up to that row:
                      .                        =  0  ->   0
                    0 0 0                      =  3  ->   3
                  0 0 . 0 0                    =  4  ->   7
                . . 0 0 0 . .                  =  3  ->  10
              0 0 0 0 . 0 0 0 0                =  8  ->  18
            . . . . 0 0 0 . . . .              =  3  ->  21
          0 0 0 0 0 0 . 0 0 0 0 0 0            = 12  ->  33
        . . . . . . 0 0 0 . . . . . .          =  3  ->  36
      0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0        = 16  ->  52
    . . . . . . . . 0 0 0 . . . . . . . .      =  3  ->  55
  0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0    = 20  ->  75
. . . . . . . . . . 0 0 0 . . . . . . . . . .  =  3  ->  78
(End)

Mathematica

rows = 53; Accumulate[Count[#, n_ /; n == 0] & /@ Table[Table[Take[CellularAutomaton[1, {{1}, 0}, rows - 1, {All, All}][[k]], {rows - k + 1, rows + k - 1}], {k, rows}][[k]], {k, 1, rows}]] (* Michael De Vlieger, Dec 14 2015 *)

Crossrefs

Cf. A265718, A265720, A265721, A267049.

Sequence in context: A042591 A098001 A171071 * A352781 A342448 A365931

Adjacent sequences: A265721 A265722 A265723 * A265725 A265726 A265727

Keyword

nonn,easy

Author

Robert Price, Dec 14 2015

Status

approved