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A118101 Decimal representation of n-th iteration of the Rule 94 elementary cellular automaton starting with a single ON cell. 3
1, 7, 27, 119, 427, 1879, 6827, 30039, 109227, 480599, 1747627, 7689559, 27962027, 123032919, 447392427, 1968526679, 7158278827, 31496426839, 114532461227, 503942829399, 1832519379627, 8063085270359, 29320310074027, 129009364325719, 469124961184427 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Rule 94

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Index entries for linear recurrences with constant coefficients, signature (0,17,0,-16).

Index entries for sequences related to cellular automata

FORMULA

a(0)=1, a(1)=7, a(n odd) = (10+11*4^n)/6, a(n even) = (1+5*4^n)/3.

From Colin Barker, Oct 08 2015: (Start)

a(n) = (12-(-4)^n-8*(-1)^n+21*4^n)/12 for n>1.

a(n) = 17*a(n-2) - 16*a(n-4) for n>5.

G.f.: -(2*x+1)*(16*x^4-5*x-1) / ((x-1)*(x+1)*(4*x-1)*(4*x+1)).

(End)

EXAMPLE

From Michael De Vlieger, Oct 08 2015: (Start)

First 8 rows, representing ON cells as "1", OFF cells within the bounds of ON cells as "0", interpreted as a binary number at left, the decimal equivalent appearing at right:

                    1 =       1

                  111 =       7

               1 1011 =      27

             111 0111 =     119

          1 1010 1011 =     427

        111 0101 0111 =   1 879

     1 1010 1010 1011 =   6 827

   111 0101 0101 0111 =  30 039

1 1010 1010 1010 1011 = 109 227

(End)

MATHEMATICA

clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; FromDigits[#, 2] & /@ Map[clip, CellularAutomaton[94, {{1}, 0}, 24]] (* or *)

{1, 7}~Join~Table[(12 - (-4)^n - 8 (-1)^n + 21*4^n)/12, {n, 2, 24}] (* Michael De Vlieger, Oct 08 2015 *)

Join[{1, 7}, LinearRecurrence[{0, 17, 0, -16}, {27, 119, 427, 1879}, 30]] (* Vincenzo Librandi, Oct 09 2015 *)

PROG

(PARI) Vec(-(2*x+1)*(16*x^4-5*x-1) / ((x-1)*(x+1)*(4*x-1)*(4*x+1)) + O(x^30)) \\ Colin Barker, Oct 08 2015

(PARI) vector(100, n, n--; if(n<2, 7^n, (12-(-4)^n-8*(-1)^n+21*4^n)/12)) \\ Altug Alkan, Oct 08 2015

(MAGMA) [1, 7] cat [(12-(-4)^n-8*(-1)^n+21*4^n) div 12: n in [2..25]]; // Vincenzo Librandi, Oct 09 2015

CROSSREFS

This sequence, A071033 and A118102 are equivalent descriptions of the Rule 94 automaton.

Sequence in context: A255278 A048711 A249184 * A201380 A269070 A147996

Adjacent sequences:  A118098 A118099 A118100 * A118102 A118103 A118104

KEYWORD

nonn,base,easy

AUTHOR

Eric W. Weisstein, Apr 12 2006

STATUS

approved

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Last modified May 4 07:27 EDT 2016. Contains 272388 sequences.