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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

A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata..., Fig. 10.

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 December 11 03:18 EST 2016. Contains 279034 sequences.