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

Author

Robert Price, Dec 14 2015

Keywords

• nonn
• easy

Examples

			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)
		

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

Crossrefs

Cf. A265718, A265720, A265721, A267049.

Programs

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 *)

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