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A265428 Number of ON (black) cells in the n-th iteration of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell. 2
1, 2, 2, 4, 4, 5, 5, 7, 7, 8, 8, 10, 10, 11, 11, 13, 13, 14, 14, 16, 16, 17, 17, 19, 19, 20, 20, 22, 22, 23, 23, 25, 25, 26, 26, 28, 28, 29, 29, 31, 31, 32, 32, 34, 34, 35, 35, 37, 37, 38, 38, 40, 40, 41, 41, 43, 43, 44, 44, 46, 46, 47, 47, 49, 49, 50, 50 (list; graph; refs; listen; history; text; internal format)
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

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*(6*n-3*(-1)^n+(1-i)*(-i)^n+(1+i)*i^n+9) where i = sqrt(-1).

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

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

a(n) = (6*n+9+2*cos(n*Pi/2)-3*cos(n*Pi)-2*sin(n*Pi/2))/8. - Wesley Ivan Hurt, Oct 02 2017

EXAMPLE

From Michael De Vlieger, Dec 09 2015: (Start)

First 12 rows, replacing "0" with ".", ignoring "0" outside of range of 1's for better visibility of ON cells, with total number of 1's in the row to the left of the chart:

1  =   1

2  =   1 1

2  =   1 . 1

4  =   1 1 1 1

4  =   1 1 1 . 1

5  =   1 1 . 1 1 1

5  =   1 . 1 1 1 . 1

7  =   1 1 1 1 . 1 1 1

7  =   1 1 1 . 1 1 1 . 1

8  =   1 1 . 1 1 1 . 1 1 1

8  =   1 . 1 1 1 . 1 1 1 . 1

10 =   1 1 1 1 . 1 1 1 . 1 1 1

10 =   1 1 1 . 1 1 1 . 1 1 1 . 1

(End)

MATHEMATICA

rule = 188; 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}]

Count[#, n_ /; n == 1] & /@ CellularAutomaton[188, {{1}, 0}, 66] (* Michael De Vlieger, Dec 09 2015 *)

PROG

(PARI) Vec((1+x+2*x^3-x^4)/((1-x)^2*(1+x)*(1+x^2)) + O(x^100)) \\ Colin Barker, Dec 14 2015

CROSSREFS

Cf. A118174.

Sequence in context: A152850 A036714 A260734 * A035644 A288773 A288774

Adjacent sequences:  A265425 A265426 A265427 * A265429 A265430 A265431

KEYWORD

nonn,easy

AUTHOR

Robert Price, Dec 08 2015

STATUS

approved

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Last modified November 21 16:01 EST 2017. Contains 295003 sequences.