%I #13 Apr 19 2024 02:00:17
%S 1,1,7,4,31,16,127,64,511,256,2047,1024,8191,4096,32767,16384,131071,
%T 65536,524287,262144,2097151,1048576,8388607,4194304,33554431,
%U 16777216,134217727,67108864,536870911,268435456,2147483647,1073741824,8589934591,4294967296
%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 475", 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="/A282453/b282453.txt">Table of n, a(n) for n = 0..126</a>
%H Robert Price, <a href="/A282453/a282453.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 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</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_, Apr 18 2024: (Start)
%F a(n) = 5*a(n-2) - 4*a(n-4) for n > 3.
%F G.f.: (-x^3 + 2*x^2 + x + 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 = 475; 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. A282451, A282452, A282454.
%K nonn,easy
%O 0,3
%A _Robert Price_, Feb 15 2017