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A284299 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 865", based on the 5-celled von Neumann neighborhood.

%I A284299 #11 Feb 16 2025 08:33:43
%S A284299 1,0,11,101,1010,11101,101010,1110111,10101011,111011101,1010101010,
%T A284299 11101110101,101010101010,1110111010101,10101010101010,
%U A284299 111011111010101,1010101110101010,11101110101110101,101010101010101010,1110111010101111101,10101010101010111010
%N A284299 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 865", based on the 5-celled von Neumann neighborhood.
%C A284299 Initialized with a single black (ON) cell at stage zero.
%D A284299 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A284299 Robert Price, <a href="/A284299/b284299.txt">Table of n, a(n) for n = 0..126</a>
%H A284299 Robert Price, <a href="/A284299/a284299.tmp.txt">Diagrams of first 20 stages</a>
%H A284299 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 A284299 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A284299 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A284299 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A284299 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A284299 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A284299 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A284299 Conjectures from _Chai Wah Wu_, May 04 2024: (Start)
%F A284299 a(n) = a(n-2) + 100000000*a(n-8) - 100000000*a(n-10) for n > 36.
%F A284299 G.f.: (100000000000000*x^36 + 100000000000*x^35 - 98990000000000*x^34 - 100000000000*x^33 - 9900000000*x^32 + 1000000000*x^31 - 100000000*x^30 + 99000000000*x^29 - 1000001000000*x^28 - 100000001000*x^27 + 989900*x^26 + 1000*x^25 + 99*x^24 - 10*x^23 + x^22 - 100990*x^21 + 10000*x^20 + 1000*x^19 - 9900000*x^17 - 990100000*x^13 + 100000000*x^12 + 890099000*x^11 - x^10 + 109900990*x^9 - 89999999*x^8 + 1099010*x^7 + 100000*x^6 + 11000*x^5 + 999*x^4 + 101*x^3 + 10*x^2 + 1)/(100000000*x^10 - 100000000*x^8 - x^2 + 1). (End)
%t A284299 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A284299 code = 865; stages = 128;
%t A284299 rule = IntegerDigits[code, 2, 10];
%t A284299 g = 2 * stages + 1; (* Maximum size of grid *)
%t A284299 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A284299 ca = a;
%t A284299 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A284299 PrependTo[ca, a];
%t A284299 (* Trim full grid to reflect growth by one cell at each stage *)
%t A284299 k = (Length[ca[[1]]] + 1)/2;
%t A284299 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 A284299 Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 10], {i, 1, stages - 1}]
%Y A284299 Cf. A284300, A284301, A284302.
%K A284299 nonn,easy
%O A284299 0,3
%A A284299 _Robert Price_, Mar 24 2017