%I #45 Sep 10 2021 22:51:47

%S 1,7,27,119,427,1879,6827,30039,109227,480599,1747627,7689559,

%T 27962027,123032919,447392427,1968526679,7158278827,31496426839,

%U 114532461227,503942829399,1832519379627,8063085270359,29320310074027,129009364325719,469124961184427

%N Decimal representation of n-th iteration of the Rule 94 elementary cellular automaton starting with a single ON cell.

%H Colin Barker, <a href="/A118101/b118101.txt">Table of n, a(n) for n = 0..1000</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. 10.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rule94.html">Rule 94</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(0)=1, a(1)=7, a(n odd) = (10+11*4^n)/6, a(n even) = (1+5*4^n)/3.

%F From _Colin Barker_, Oct 08 2015 and Apr 16 2019: (Start)

%F a(n) = (12-(-4)^n-8*(-1)^n+21*4^n)/12 for n>1.

%F a(n) = 17*a(n-2) - 16*a(n-4) for n>5.

%F G.f.: -(2*x+1)*(16*x^4-5*x-1) / ((x-1)*(x+1)*(4*x-1)*(4*x+1)).

%F (End)

%e From _Michael De Vlieger_, Oct 08 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 1011 = 27

%e 111 0111 = 119

%e 1 1010 1011 = 427

%e 111 0101 0111 = 1 879

%e 1 1010 1010 1011 = 6 827

%e 111 0101 0101 0111 = 30 039

%e 1 1010 1010 1010 1011 = 109 227

%e (End)

%t clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; FromDigits[#, 2] & /@ Map[clip, CellularAutomaton[94, {{1}, 0}, 24]] (* _Michael De Vlieger_, Oct 08 2015 *)

%o (Python) print([(11*4**n + 10)//6 - 2*0**abs(n-1) if n%2 else (5*4**n + 1)//3 - 0**n for n in range(50)]) # _Karl V. Keller, Jr._, Sep 10 2021

%Y This sequence, A071033 and A118102 are equivalent descriptions of the Rule 94 automaton.

%K nonn,base,easy

%O 0,2

%A _Eric W. Weisstein_, Apr 12 2006