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%I #11 Jun 15 2020 01:50:05
%S 1,0,3,10,0,63,0,255,0,1023,0,4095,0,16383,0,65535,0,262143,0,1048575,
%T 0,4194303,0,16777215,0,67108863,0,268435455,0,1073741823,0,
%U 4294967295,0,17179869183,0,68719476735,0,274877906943,0,1099511627775,0,4398046511103,0
%N Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 73", based on the 5-celled von Neumann neighborhood.
%C Initialized with a single black (ON) cell at stage zero.
%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H Robert Price, <a href="/A278759/b278759.txt">Table of n, a(n) for n = 0..126</a>
%H Robert Price, <a href="/A278759/a278759.tmp.txt">Diagrams of first 20 stages</a>
%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F Conjectures from _Chai Wah Wu_, Jun 14 2020: (Start)
%F a(n) = 5*a(n-2) - 4*a(n-4) for n > 7.
%F G.f.: (-20*x^7 + 12*x^6 + 13*x^5 - 11*x^4 + 10*x^3 - 2*x^2 + 1)/(4*x^4 - 5*x^2 + 1). (End)
%t CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t code=73; stages=128;
%t rule=IntegerDigits[code,2,10];
%t g=2*stages+1; (* Maximum size of grid *)
%t a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t ca=a;
%t ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t PrependTo[ca,a];
%t (* Trim full grid to reflect growth by one cell at each stage *)
%t k=(Length[ca[[1]]]+1)/2;
%t ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];
%t Table[FromDigits[Part[ca[[i]][[i]],Range[1,i]],2], {i,1,stages-1}]
%Y Cf. A278757, A278758, A278760.
%K nonn,easy
%O 0,3
%A _Robert Price_, Nov 27 2016
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