A118108 Decimal representation of n-th iteration of the Rule 54 elementary cellular automaton starting with a single black cell.

1, 7, 17, 119, 273, 1911, 4369, 30583, 69905, 489335, 1118481, 7829367, 17895697, 125269879, 286331153, 2004318071, 4581298449, 32069089143, 73300775185, 513105426295, 1172812402961, 8209686820727, 18764998447377, 131354989131639, 300239975158033

Offset

0,2

Comments

a(1660) is 1000 digits long. - Michael De Vlieger, Oct 07 2015

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1660

A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata..., Fig 8.

Eric Weisstein's World of Mathematics, Rule 54

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Index entries for sequences related to cellular automata

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

Formula

a(n) = 7*(4^(n+1)-1)/15 for n odd; a(n) = (4^(n+2)-1)/15 for n even.

From Colin Barker, Oct 08 2015 and Apr 16 2019: (Start)

a(n) = 17*a(n-2) - 16*a(n-4) for n>3.

G.f.: (7*x+1) / ((x-1)*(x+1)*(4*x-1)*(4*x+1)).

(End)

a(n) = floor((16+12*(n mod 2))*4^n/15). - Karl V. Keller, Jr., Aug 04 2021

Example

From Michael De Vlieger, Oct 07 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 0001 =    17
             111 0111 =   119
          1 0001 0001 =   273
        111 0111 0111 =  1911
     1 0001 0001 0001 =  4369
   111 0111 0111 0111 = 30583
1 0001 0001 0001 0001 = 69905
(End)

Mathematica

clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; FromDigits[#, 2] & /@ Map[clip, CellularAutomaton[54, {{1}, 0}, 27]] (* or *)
Table[If[EvenQ@ n, (4^(n + 2) - 1), 7 (4^(n + 1) - 1)]/15, {n, 0, 27}] (* Michael De Vlieger, Oct 07 2015 *)

Prog

(Python) print([(16+12*(n%2))*4**n//15 for n in range(30)]) # Karl V. Keller, Jr., Aug 04 2021

Crossrefs

See A071030, A118109 for two other versions of this sequence.

Sequence in context: A063384 A165246 A266382 * A227506 A244279 A364703

Adjacent sequences: A118105 A118106 A118107 * A118109 A118110 A118111

Keyword

nonn,base,easy

Author

Eric W. Weisstein, Apr 13 2006

Extensions

a(23)-a(24) from Michael De Vlieger, Oct 07 2015

Status

approved