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A266069 Decimal representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell. 1
1, 4, 2, 121, 4, 2035, 8, 32743, 16, 524239, 32, 8388511, 64, 134217535, 128, 2147483263, 256, 34359737599, 512, 549755812351, 1024, 8796093019135, 2048, 140737488349183, 4096, 2251799813672959, 8192, 36028797018939391, 16384, 576460752303374335, 32768 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Rule 35 also generates this sequence.

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 (0,19,0,-50,0,32).

FORMULA

G.f.: (1+4*x-17*x^2+45*x^3+16*x^4-64*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)). - Colin Barker, Dec 21 2015

a(n) = ((sqrt(2)+3)*((-1)^n+1)-6)*sqrt(2)^(n-3) - (2*4^n-1)*((-1)^n-1)/2. Therefore: a(n) = 2^(n/2) for even n; otherwise, a(n) = 2^(2*n+1)-3*2^((n-1)/2)-1. [Bruno Berselli, Dec 21 2015]

EXAMPLE

From Michael De Vlieger, Dec 21 2015: (Start)

First 8 rows, replacing leading zeros with ".", the row converted to its binary, then decimal equivalent at right:

              1                =               1 ->     1

            1 0 0              =             100 ->     4

          . . . 1 0            =              10 ->     2

        1 1 1 1 0 0 1          =         1111001 ->   121

      . . . . . . 1 0 0        =             100 ->     4

    1 1 1 1 1 1 1 0 0 1 1      =     11111110011 ->  2035

  . . . . . . . . . 1 0 0 0    =            1000 ->     8

1 1 1 1 1 1 1 1 1 1 0 0 1 1 1  = 111111111100111 -> 32743

(End)

MATHEMATICA

rule = 3; rows = 30; Table[FromDigits[Table[Take[CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}][[k]], {rows-k+1, rows+k-1}], {k, 1, rows}][[k]], 2], {k, 1, rows}]

Table[((Sqrt[2] + 3) ((-1)^n + 1) - 6) Sqrt[2]^(n - 3) - (2 4^n - 1) ((-1)^n - 1)/2, {n, 0, 30}] (* Bruno Berselli, Dec 22 2015 *)

LinearRecurrence[{0, 19, 0, -50, 0, 32}, {1, 4, 2, 121, 4, 2035}, 40] (* Vincenzo Librandi, Dec 22 2015 *)

PROG

(PARI) Vec((1+4*x-17*x^2+45*x^3+16*x^4-64*x^5)/((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)) + O(x^40)) \\ Colin Barker, Dec 21 2015

(MAGMA) [IsEven(n) select 2^(n div 2) else 2^(2*n+1)-3*2^((n-1) div 2)-1: n in [0..30]]; // Bruno Berselli, Dec 22 2015

(Sage) [((sqrt(2)+3)*((-1)^n+1)-6)*sqrt(2)^(n-3)-(2*4^n-1)*((-1)^n-1)/2 for n in (0..30)] # Bruno Berselli, Dec 22 2015

CROSSREFS

Cf. A263428, A266068, A266070, A266071, A081253, A266072, A247375, A266073, A266074.

Sequence in context: A236381 A201228 A010319 * A057167 A096683 A158903

Adjacent sequences:  A266066 A266067 A266068 * A266070 A266071 A266072

KEYWORD

nonn,easy

AUTHOR

Robert Price, Dec 20 2015

STATUS

approved

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Last modified November 16 23:51 EST 2018. Contains 317275 sequences. (Running on oeis4.)