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

%I #12 May 06 2024 15:19:26

%S 1,2,4,9,18,36,75,145,300,579,1213,2334,4804,9273,19410,37348,76883,

%T 148393,310532,597627,1230097,2374372,4968531,9561889,19681356,

%U 37990195,79496909,152990002,314902220,607844659,1271953101,2447845682,5038429900,9725511987

%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 541", 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="/A282986/b282986.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A282986/a282986.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) = 17*a(n-4) - 16*a(n-8) for n > 33.

%F G.f.: (-4096*x^33 - 6144*x^32 + 5872*x^31 + 2144*x^30 + 1296*x^29 + 720*x^28 - 79*x^27 + 394*x^26 + 159*x^25 - 165*x^24 - 202*x^23 + 47*x^22 + 59*x^21 - 50*x^20 + 55*x^19 - 30*x^18 + 16*x^17 + 15*x^16 - 10*x^15 - 11*x^14 + 6*x^13 - 8*x^12 + 13*x^11 + 2*x^10 - x^9 + 10*x^8 - 8*x^7 + 7*x^6 + 2*x^5 + x^4 + 9*x^3 + 4*x^2 + 2*x + 1)/(16*x^8 - 17*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 = 541; 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. A282984, A282985, A282987.

%K nonn,easy

%O 0,2

%A _Robert Price_, Feb 26 2017