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Decimal representation of the x-axis, from the left edge to the origin, (and also from the origin to the right edge) of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 209", based on the 5-celled von Neumann neighborhood.
5

%I #19 Aug 02 2021 02:39:04

%S 1,0,7,0,31,0,127,0,511,0,2047,0,8191,0,32767,0,131071,0,524287,0,

%T 2097151,0,8388607,0,33554431,0,134217727,0,536870911,0,2147483647,0,

%U 8589934591,0,34359738367,0,137438953471,0,549755813887,0,2199023255551,0

%N Decimal representation of the x-axis, from the left edge to the origin, (and also from the origin to the right edge) of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 209", based on the 5-celled von Neumann neighborhood.

%C Initialized with a single black (ON) cell at stage zero.

%C The nonzero bisection appears to be A083420. - _Tom Copeland_, Dec 27 2016

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

%H Robert Price, <a href="/A279872/b279872.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A279872/a279872.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_, Aug 02 2021: (Start)

%F a(n) = 5*a(n-2) - 4*a(n-4) for n > 3.

%F G.f.: (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 = 209; 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. A279118, A083420.

%K nonn,easy

%O 0,3

%A _Robert Price_, Dec 21 2016