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%I #13 Jun 15 2020 21:10:02
%S 1,0,11,1000,111,110000,1111,11100000,11111,1111000000,111111,
%T 111110000000,1111111,11111100000000,11111111,1111111000000000,
%U 111111111,111111110000000000,1111111111,11111111100000000000,11111111111,1111111111000000000000,111111111111
%N Binary 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 65", 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="/A278753/b278753.txt">Table of n, a(n) for n = 0..126</a>
%H Robert Price, <a href="/A278753/a278753.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 15 2020: (Start)
%F a(n) = 111*a(n-2) - 1110*a(n-4) + 1000*a(n-6) for n > 5.
%F G.f.: (1000*x^5 - 1000*x^3 + 100*x^2 - 1)/(1000*x^6 - 1110*x^4 + 111*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=65; 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]],10], {i,1,stages-1}]
%Y Cf. A278754, A278755, A278756.
%K nonn,easy
%O 0,3
%A _Robert Price_, Nov 27 2016
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