login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 9 06:44 EST 2016. Contains 278963 sequences.