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A284300 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 865", based on the 5-celled von Neumann neighborhood.

%I A284300 #11 Feb 16 2025 08:33:43
%S A284300 1,0,110,1010,1010,101110,101010,11101110,110101010,1011101110,
%T A284300 1010101010,101011101110,101010101010,10101011101110,10101010101010,
%U A284300 1010101111101110,1010101110101010,101011101011101110,101010101010101010,10111110101011101110
%N A284300 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 865", based on the 5-celled von Neumann neighborhood.
%C A284300 Initialized with a single black (ON) cell at stage zero.
%D A284300 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A284300 Robert Price, <a href="/A284300/b284300.txt">Table of n, a(n) for n = 0..126</a>
%H A284300 Robert Price, <a href="/A284300/a284300.tmp.txt">Diagrams of first 20 stages</a>
%H A284300 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 A284300 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A284300 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A284300 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A284300 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A284300 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A284300 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A284300 Conjectures from _Chai Wah Wu_, May 04 2024: (Start)
%F A284300 a(n) = 100*a(n-2) + a(n-8) - 100*a(n-10) for n > 36.
%F A284300 G.f.: (10000000000000000000000*x^36 + 1000000000000000000000000*x^35 + 1009900000000000000000000*x^34 - 10000000000000000000000*x^33 + 990000000000000000000000*x^32 + 10000000000000000000000*x^31 - 10000000000000000000000*x^30 - 99000000000000000000*x^29 - 10000010000000000000000*x^28 - 1000000010000000000000000*x^27 - 1009900000000000000000000*x^26 + 10000000000000000000000*x^25 - 990000000000000000000000*x^24 - 10000000000000000000000*x^23 + 10000000000000000000000*x^22 + 98990000000000000000*x^21 + 10000000000000000*x^20 + 10000000000000000*x^19 + 990000000000*x^17 - 99010000*x^13 + 10000*x^12 - 99010900*x^11 - 10000000000*x^10 - 99009890*x^9 + 100000009*x^8 + 990110*x^7 + 10*x^6 + 110*x^5 - 9990*x^4 + 1010*x^3 + 10*x^2 + 1)/(100*x^10 - x^8 - 100*x^2 + 1). (End)
%t A284300 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A284300 code = 865; stages = 128;
%t A284300 rule = IntegerDigits[code, 2, 10];
%t A284300 g = 2 * stages + 1; (* Maximum size of grid *)
%t A284300 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A284300 ca = a;
%t A284300 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A284300 PrependTo[ca, a];
%t A284300 (* Trim full grid to reflect growth by one cell at each stage *)
%t A284300 k = (Length[ca[[1]]] + 1)/2;
%t A284300 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 A284300 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A284300 Cf. A284299, A284301, A284302.
%K A284300 nonn,easy
%O A284300 0,3
%A A284300 _Robert Price_, Mar 24 2017