cp's OEIS Frontend

A290195 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.

%I A290195 #14 Feb 16 2025 08:33:49
%S A290195 1,1,5,11,7,47,31,191,127,767,511,3071,2047,12287,8191,49151,32767,
%T A290195 196607,131071,786431,524287,3145727,2097151,12582911,8388607,
%U A290195 50331647,33554431,201326591,134217727,805306367,536870911,3221225471,2147483647,12884901887
%N A290195 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.
%C A290195 Initialized with a single black (ON) cell at stage zero.
%D A290195 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A290195 Robert Price, <a href="/A290195/b290195.txt">Table of n, a(n) for n = 0..126</a>
%H A290195 Robert Price, <a href="/A290195/a290195.tmp.txt">Diagrams of first 20 stages</a>
%H A290195 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 A290195 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A290195 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A290195 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A290195 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A290195 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A290195 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A290195 Conjecture: For odd n > 3, a(n) = 2^(n-1) - 1, for even n > 3, a(n) = 3*2^(n-1) - 1. - _David A. Corneth_, Jul 23 2017
%F A290195 From _Chai Wah Wu_, Nov 01 2018: (Start)
%F A290195 a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n > 5 (conjectured).
%F A290195 G.f.: (16*x^5 - 20*x^4 + 6*x^3 + 1)/((x - 1)*(2*x - 1)*(2*x + 1)) (conjectured). (End)
%t A290195 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A290195 code = 705; stages = 128;
%t A290195 rule = IntegerDigits[code, 2, 10];
%t A290195 g = 2 * stages + 1; (* Maximum size of grid *)
%t A290195 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A290195 ca = a;
%t A290195 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A290195 PrependTo[ca, a];
%t A290195 (* Trim full grid to reflect growth by one cell at each stage *)
%t A290195 k = (Length[ca[[1]]] + 1)/2;
%t A290195 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 A290195 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A290195 Cf. A290192, A290193, A290194.
%K A290195 nonn,easy
%O A290195 0,3
%A A290195 _Robert Price_, Jul 23 2017