A265431 Total number of OFF (white) cells after n iterations of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.

0, 1, 4, 7, 12, 18, 26, 34, 44, 55, 68, 81, 96, 112, 130, 148, 168, 189, 212, 235, 260, 286, 314, 342, 372, 403, 436, 469, 504, 540, 578, 616, 656, 697, 740, 783, 828, 874, 922, 970, 1020, 1071, 1124, 1177, 1232, 1288, 1346, 1404, 1464, 1525, 1588, 1651

Offset

0,3

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 09 2015 and Apr 16 2019: (Start)

a(n) = 1/16*(10*n^2+8*n+3*(-1)^n-2*(-i)^n-2*i^n+1) where i = sqrt(-1).

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

(End)

Example

From Michael De Vlieger, Dec 09 2015: (Start)
First 12 rows, replacing "1" 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                      =  1 ->  1
                  . 0 . 0 0                    =  3 ->  4
                . . . . 0 0 0                  =  3 ->  7
              . . . 0 . 0 0 0 0                =  5 -> 12
            . . 0 . . . 0 0 0 0 0              =  6 -> 18
          . 0 . . . 0 . 0 0 0 0 0 0            =  8 -> 26
        . . . . 0 . . . 0 0 0 0 0 0 0          =  8 -> 34
      . . . 0 . . . 0 . 0 0 0 0 0 0 0 0        = 10 -> 44
    . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0      = 11 -> 55
  . 0 . . . 0 . . . 0 . 0 0 0 0 0 0 0 0 0 0    = 13 -> 68
. . . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0 0 0  = 13 -> 81
(End)

Mathematica

lim = 104; a = {}; Do[AppendTo[a, Take[#[[k]], 2 (k - 1) + 1]], {k, Floor[Length[#]/2]}] &@ CellularAutomaton[188, {{1}, 0}, lim]; Accumulate[Count[#, n_ /; n == 0] & /@ a] (* Michael De Vlieger, Dec 09 2015 *)

Crossrefs

Cf. A118174.

Sequence in context: A336660 A310792 A178907 * A132297 A007333 A097536

Adjacent sequences: A265428 A265429 A265430 * A265432 A265433 A265434

Keyword

nonn,easy

Author

Robert Price, Dec 08 2015

Status

approved