cp's OEIS Frontend

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

%I A118108 #45 Feb 16 2025 08:33:00
%S A118108 1,7,17,119,273,1911,4369,30583,69905,489335,1118481,7829367,17895697,
%T A118108 125269879,286331153,2004318071,4581298449,32069089143,73300775185,
%U A118108 513105426295,1172812402961,8209686820727,18764998447377,131354989131639,300239975158033
%N A118108 Decimal representation of n-th iteration of the Rule 54 elementary cellular automaton starting with a single black cell.
%C A118108 a(1660) is 1000 digits long. - _Michael De Vlieger_, Oct 07 2015
%H A118108 Michael De Vlieger, <a href="/A118108/b118108.txt">Table of n, a(n) for n = 0..1660</a>
%H A118108 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 A118108 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule54.html">Rule 54</a>
%H A118108 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A118108 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A118108 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,17,0,-16).
%F A118108 a(n) = 7*(4^(n+1)-1)/15 for n odd; a(n) = (4^(n+2)-1)/15 for n even.
%F A118108 From _Colin Barker_, Oct 08 2015 and Apr 16 2019: (Start)
%F A118108 a(n) = 17*a(n-2) - 16*a(n-4) for n>3.
%F A118108 G.f.: (7*x+1) / ((x-1)*(x+1)*(4*x-1)*(4*x+1)).
%F A118108 (End)
%F A118108 a(n) = floor((16+12*(n mod 2))*4^n/15). - _Karl V. Keller, Jr._, Aug 04 2021
%e A118108 From _Michael De Vlieger_, Oct 07 2015: (Start)
%e A118108 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 A118108                     1 =     1
%e A118108                   111 =     7
%e A118108                1 0001 =    17
%e A118108              111 0111 =   119
%e A118108           1 0001 0001 =   273
%e A118108         111 0111 0111 =  1911
%e A118108      1 0001 0001 0001 =  4369
%e A118108    111 0111 0111 0111 = 30583
%e A118108 1 0001 0001 0001 0001 = 69905
%e A118108 (End)
%t A118108 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 A118108 Table[If[EvenQ@ n, (4^(n + 2) - 1), 7 (4^(n + 1) - 1)]/15, {n, 0, 27}] (* _Michael De Vlieger_, Oct 07 2015 *)
%o A118108 (Python) print([(16+12*(n%2))*4**n//15 for n in range(30)]) # _Karl V. Keller, Jr._, Aug 04 2021
%Y A118108 See A071030, A118109 for two other versions of this sequence.
%K A118108 nonn,base,easy
%O A118108 0,2
%A A118108 _Eric W. Weisstein_, Apr 13 2006
%E A118108 a(23)-a(24) from _Michael De Vlieger_, Oct 07 2015