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Decimal 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 734", based on the 5-celled von Neumann neighborhood.
4

%I #10 May 06 2024 15:46:35

%S 1,3,7,15,29,63,119,255,477,1023,1915,4091,7647,16383,30647,65535,

%T 122367,262143,490367,1048575,1957631,4194303,7847935,16777215,

%U 31326207,67108863,125566975,268435455,501219327,1073741823,2009071615,4294967295,8019509247

%N Decimal 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 734", 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="/A283864/b283864.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A283864/a283864.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 06 2024: (Start)

%F a(n) = a(n-1) + 16*a(n-4) - 16*a(n-5) for n > 25.

%F G.f.: (-4096*x^25 + 4096*x^24 - 2048*x^23 + 2048*x^22 + 256*x^21 - 256*x^20 - 64*x^16 + 72*x^15 - 8*x^14 + 4*x^12 - 4*x^10 + 2*x^9 - 2*x^8 + 8*x^7 - 8*x^6 + 2*x^5 - 2*x^4 + 8*x^3 + 4*x^2 + 2*x + 1)/(16*x^5 - 16*x^4 - x + 1). (End)

%t CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];

%t code = 734; 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. A283849, A283850, A283865.

%K nonn,easy

%O 0,2

%A _Robert Price_, Mar 17 2017