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A265429 Total number of ON (black) cells after n iterations of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell. 2
1, 3, 5, 9, 13, 18, 23, 30, 37, 45, 53, 63, 73, 84, 95, 108, 121, 135, 149, 165, 181, 198, 215, 234, 253, 273, 293, 315, 337, 360, 383, 408, 433, 459, 485, 513, 541, 570, 599, 630, 661, 693, 725, 759, 793, 828, 863, 900, 937, 975, 1013, 1053, 1093, 1134 (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 (2,-1,0,1,-2,1).

FORMULA

From Colin Barker, Dec 09 2015: (Start)

a(n) = 1/16*(6*n^2+24*n-3*(-1)^n+2*(-i)^n+2*i^n+15) where i = sqrt(-1).

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

(End)

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, followed by total number of ON cells per row, and running total up to that row:

1                          =  1 ->   1

1 1                        =  2 ->   3

1 . 1                      =  2 ->   5

1 1 1 1                    =  4 ->   9

1 1 1 . 1                  =  4 ->  13

1 1 . 1 1 1                =  5 ->  18

1 . 1 1 1 . 1              =  5 ->  23

1 1 1 1 . 1 1 1            =  7 ->  30

1 1 1 . 1 1 1 . 1          =  7 ->  37

1 1 . 1 1 1 . 1 1 1        =  8 ->  45

1 . 1 1 1 . 1 1 1 . 1      =  8 ->  53

1 1 1 1 . 1 1 1 . 1 1 1    = 10 ->  63

1 1 1 . 1 1 1 . 1 1 1 . 1  = 10 ->  72

(End)

MATHEMATICA

rule = 188; rows = 30; Table[Total[Take[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}], k]], {k, 1, rows}]

Accumulate[Count[#, n_ /; n == 1] & /@ CellularAutomaton[188, {{1}, 0}, 53]] (* Michael De Vlieger, Dec 09 2015 *)

PROG

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

CROSSREFS

Cf. A118174.

Sequence in context: A032635 A036713 A260733 * A122248 A024403 A129230

Adjacent sequences:  A265426 A265427 A265428 * A265430 A265431 A265432

KEYWORD

nonn,easy

AUTHOR

Robert Price, Dec 08 2015

STATUS

approved

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Last modified September 25 16:25 EDT 2017. Contains 292499 sequences.