cp's OEIS Frontend

A279872 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.

%I A279872 #20 Feb 16 2025 08:33:38
%S A279872 1,0,7,0,31,0,127,0,511,0,2047,0,8191,0,32767,0,131071,0,524287,0,
%T A279872 2097151,0,8388607,0,33554431,0,134217727,0,536870911,0,2147483647,0,
%U A279872 8589934591,0,34359738367,0,137438953471,0,549755813887,0,2199023255551,0
%N A279872 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 A279872 Initialized with a single black (ON) cell at stage zero.
%C A279872 The nonzero bisection appears to be A083420. - _Tom Copeland_, Dec 27 2016
%D A279872 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A279872 Robert Price, <a href="/A279872/b279872.txt">Table of n, a(n) for n = 0..126</a>
%H A279872 Robert Price, <a href="/A279872/a279872.tmp.txt">Diagrams of first 20 stages</a>
%H A279872 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 A279872 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A279872 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A279872 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A279872 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A279872 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A279872 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A279872 Conjectures from _Chai Wah Wu_, Aug 02 2021: (Start)
%F A279872 a(n) = 5*a(n-2) - 4*a(n-4) for n > 3.
%F A279872 G.f.: (2*x^2 + 1)/(4*x^4 - 5*x^2 + 1). (End)
%t A279872 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A279872 code = 209; stages = 128;
%t A279872 rule = IntegerDigits[code, 2, 10];
%t A279872 g = 2 * stages + 1; (* Maximum size of grid *)
%t A279872 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A279872 ca = a;
%t A279872 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A279872 PrependTo[ca, a];
%t A279872 (* Trim full grid to reflect growth by one cell at each stage *)
%t A279872 k = (Length[ca[[1]]] + 1)/2;
%t A279872 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 A279872 Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 2], {i, 1, stages - 1}]
%Y A279872 Cf. A279118, A083420.
%K A279872 nonn,easy
%O A279872 0,3
%A A279872 _Robert Price_, Dec 21 2016