%I #10 Feb 16 2025 08:33:40
%S 1,3,6,15,26,63,106,255,426,1023,1706,4095,6826,16383,27306,65535,
%T 109226,262143,436906,1048575,1747626,4194303,6990506,16777215,
%U 27962026,67108863,111848106,268435455,447392426,1073741823,1789569706,4294967295,7158278826
%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 430", 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="/A282123/b282123.txt">Table of n, a(n) for n = 0..126</a>
%H Robert Price, <a href="/A282123/a282123.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="https://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 _Colin Barker_, Feb 07 2017: (Start)
%F a(n) = (-5 - (-2)^n + (-1)^n + 11*2^n) / 6.
%F a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
%F G.f.: (1 + 3*x + x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
%F (End)
%t CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t code = 430; 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. A282121, A282122, A282124.
%K nonn,easy,changed
%O 0,2
%A _Robert Price_, Feb 06 2017