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A265430 Number of OFF (white) cells in the n-th iteration of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell. 2
0, 1, 3, 3, 5, 6, 8, 8, 10, 11, 13, 13, 15, 16, 18, 18, 20, 21, 23, 23, 25, 26, 28, 28, 30, 31, 33, 33, 35, 36, 38, 38, 40, 41, 43, 43, 45, 46, 48, 48, 50, 51, 53, 53, 55, 56, 58, 58, 60, 61, 63, 63, 65, 66, 68, 68, 70, 71, 73, 73, 75, 76, 78, 78, 80, 81, 83 (list; graph; refs; listen; history; text; internal format)
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

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

FORMULA

From Colin Barker, Dec 09 2015: (Start)

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

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

G.f.: x*(1+2*x+2*x^3) / ((1-x)^2*(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, with total number of 0's in the row to the right of the chart:

                      .                        =  0

                    . . 0                      =  1

                  . 0 . 0 0                    =  3

                . . . . 0 0 0                  =  3

              . . . 0 . 0 0 0 0                =  5

            . . 0 . . . 0 0 0 0 0              =  6

          . 0 . . . 0 . 0 0 0 0 0 0            =  8

        . . . . 0 . . . 0 0 0 0 0 0 0          =  8

      . . . 0 . . . 0 . 0 0 0 0 0 0 0 0        = 10

    . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0      = 11

  . 0 . . . 0 . . . 0 . 0 0 0 0 0 0 0 0 0 0    = 13

. . . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0 0 0  = 13

(End)

MATHEMATICA

lim = 134; a = {}; Do[AppendTo[a, Take[#[[k]], 2 (k - 1) + 1]], {k, Floor[Length[#]/2]}] &@ CellularAutomaton[188, {{1}, 0}, lim]; Count[#, n_ /; n == 0] & /@ a (* Michael De Vlieger, Dec 09 2015 *)

PROG

(PARI) concat(0, Vec(x*(1+2*x+2*x^3)/((1-x)^2*(1+x)*(1+x^2)) + O(x^100))) \\ Colin Barker, Dec 14 2015

CROSSREFS

Cf. A118174.

Sequence in context: A116592 A152772 A089175 * A262355 A138373 A011976

Adjacent sequences:  A265427 A265428 A265429 * A265431 A265432 A265433

KEYWORD

nonn,easy

AUTHOR

Robert Price, Dec 08 2015

STATUS

approved

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Last modified September 23 13:55 EDT 2017. Contains 292358 sequences.