%I #11 May 05 2024 19:28:55
%S 1,10,100,1001,10010,100100,1001011,10010001,100101100,1001000011,
%T 10010111101,100100010010,1001011000000,10010000111111,
%U 100101111011101,1001000100101010,10010110011010000,100100001000100111,1001011110111011001,10010001001010100110
%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 533", 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="/A282954/b282954.txt">Table of n, a(n) for n = 0..126</a>
%H Robert Price, <a href="/A282954/a282954.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_, May 05 2024: (Start)
%F a(n) = 10001*a(n-4) - 10000*a(n-8) for n > 24.
%F G.f.: (990000000*x^24 - 991000000*x^23 + 1000000*x^22 + 110000000*x^21 - 9999000*x^20 - 900*x^19 - 100*x^18 - 111011000*x^17 + 11010000*x^16 + 1000*x^15 + 1100*x^13 - 1100*x^12 + 9*x^11 + 90*x^10 - 89*x^9 + 1090*x^8 - 1000*x^7 + 911*x^6 + 90*x^5 + 9*x^4 + 1001*x^3 + 100*x^2 + 10*x + 1)/(10000*x^8 - 10001*x^4 + 1). (End)
%t CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t code = 533; 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. A282955, A282956, A282957.
%K nonn,easy
%O 0,2
%A _Robert Price_, Feb 25 2017
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