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 A118108 Decimal representation of n-th iteration of the Rule 54 elementary cellular automaton starting with a single black cell. 6

%I

%S 1,7,17,119,273,1911,4369,30583,69905,489335,1118481,7829367,17895697,

%T 125269879,286331153,2004318071,4581298449,32069089143,73300775185,

%U 513105426295,1172812402961,8209686820727,18764998447377,131354989131639,300239975158033

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

%C a(1660) is 1000 digits long. - _Michael De Vlieger_, Oct 07 2015

%H Michael De Vlieger, <a href="/A118108/b118108.txt">Table of n, a(n) for n = 0..1660</a>

%H A. J. Macfarlane, <a href="http://www.damtp.cam.ac.uk/user/ajm/Papers2016/GFsForCAsOfEvenRuleNo.ps">Generating functions for integer sequences defined by the evolution of cellular automata...</a>, Fig 8.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rule54.html">Rule 54</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,17,0,-16).

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

%F From _Colin Barker_, Oct 08 2015: (Start)

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

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

%F (End)

%e From _Michael De Vlieger_, Oct 07 2015: (Start)

%e 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:

%e 1 = 1

%e 111 = 7

%e 1 0001 = 17

%e 111 0111 = 119

%e 1 0001 0001 = 273

%e 111 0111 0111 = 1911

%e 1 0001 0001 0001 = 4369

%e 111 0111 0111 0111 = 30583

%e 1 0001 0001 0001 0001 = 69905

%e (End)

%t clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; FromDigits[#, 2] & /@ Map[clip, CellularAutomaton[54, {{1}, 0}, 27]] (* or *)

%t Table[If[EvenQ@ n, (4^(n + 2) - 1), 7 (4^(n + 1) - 1)]/15, {n, 0, 27}] (* _Michael De Vlieger_, Oct 07 2015 *)

%o (PARI) Vec((7*x+1)/((x-1)*(x+1)*(4*x-1)*(4*x+1)) + O(x^30)) \\ _Colin Barker_, Oct 08 2015

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

%K nonn,tabf,base,easy

%O 0,2

%A _Eric W. Weisstein_, Apr 13 2006

%E a(23)-a(24) from _Michael De Vlieger_, Oct 07 2015

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