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%I #11 May 06 2024 16:24:04
%S 1,3,7,13,25,59,127,215,407,957,2029,3429,6437,15245,32665,55067,
%T 103967,245271,518935,876821,1646865,3899667,8355843,14090241,
%U 26607617,62783491,132841475,224460801,421593089,998309891,2139095043,3607101441,6811549697
%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 782", 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="/A284026/b284026.txt">Table of n, a(n) for n = 0..126</a>
%H Robert Price, <a href="/A284026/a284026.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) - a(n-2) + a(n-3) + 256*a(n-8) - 256*a(n-9) + 256*a(n-10) - 256*a(n-11) for n > 32.
%F G.f.: (4096*x^32 + 5120*x^30 + 1024*x^28 + 2560*x^27 - 1024*x^26 + 1536*x^25 - 4368*x^24 - 2560*x^23 - 3348*x^22 - 2560*x^21 - 772*x^20 - 10*x^19 - 1020*x^18 + 378*x^17 - 240*x^16 - 22*x^15 - 52*x^14 - 32*x^13 - 16*x^12 - 98*x^11 - 16*x^10 + 126*x^9 + 4*x^8 + 122*x^7 + 80*x^6 + 40*x^5 + 16*x^4 + 8*x^3 + 5*x^2 + 2*x + 1)/(256*x^11 - 256*x^10 + 256*x^9 - 256*x^8 - x^3 + x^2 - 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 = 782; 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. A284024, A284025, A284027.
%K nonn,easy
%O 0,2
%A _Robert Price_, Mar 18 2017
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