%I #12 Feb 16 2025 08:33:42
%S 1,1,1,9,9,9,105,137,105,777,1513,1161,105,16137,24041,21641,5737,
%T 233737,318953,414857,218729,3739913,5103081,6624393,3528297,59838729,
%U 81649129,105976969,56481385,957419785,1306385897,1695618185,903730793,15318716681
%N Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 533", 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="/A282957/b282957.txt">Table of n, a(n) for n = 0..126</a>
%H Robert Price, <a href="/A282957/a282957.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 _Chai Wah Wu_, May 05 2024: (Start)
%F a(n) = 17*a(n-4) - 16*a(n-8) for n > 24.
%F G.f.: (-98304*x^24 - 81920*x^23 + 65536*x^22 + 24576*x^21 + 122880*x^20 + 65536*x^19 - 65536*x^18 - 28160*x^17 + 5632*x^16 + 4096*x^15 + 3072*x^13 - 1536*x^12 - 1024*x^11 - 256*x^10 + 640*x^9 - 32*x^8 - 16*x^7 + 88*x^6 - 8*x^5 - 8*x^4 + 9*x^3 + x^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 = 533; 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[i, 2 * i - 1]], 2], {i ,1, stages - 1}]
%Y Cf. A282954, A282955, A282956.
%K nonn,easy,changed
%O 0,4
%A _Robert Price_, Feb 25 2017