cp's OEIS Frontend

A290194 Decimal representation of the diagonal from the corner to the origin 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 A290194 #15 Feb 16 2025 08:33:49
%S A290194 1,2,5,13,28,61,124,253,508,1021,2044,4093,8188,16381,32764,65533,
%T A290194 131068,262141,524284,1048573,2097148,4194301,8388604,16777213,
%U A290194 33554428,67108861,134217724,268435453,536870908,1073741821,2147483644,4294967293,8589934588
%N A290194 Decimal representation of the diagonal from the corner to the origin 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 A290194 Initialized with a single black (ON) cell at stage zero.
%D A290194 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A290194 Robert Price, <a href="/A290194/b290194.txt">Table of n, a(n) for n = 0..126</a>
%H A290194 Robert Price, <a href="/A290194/a290194.tmp.txt">Diagrams of first 20 stages</a>
%H A290194 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 A290194 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A290194 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A290194 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A290194 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A290194 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A290194 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A290194 Conjecture: a(n) = Fibonacci(2*n+1) if n <= 3, for n > 3, a(n) = 2*a(n-1) + 2 if n is even, a(n) = 2*a(n-1) + 5 if n is odd. It would follow that a(n) = 2^(n+1) - 4 + (n mod 2) for n >= 3. - _David A. Corneth_, Jul 23 2017
%F A290194 From _Chai Wah Wu_, Nov 01 2018: (Start)
%F A290194 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n > 5 (conjectured).
%F A290194 G.f.: (2*x^5 + x^4 + 3*x^3 + 1)/((x - 1)*(x + 1)*(2*x - 1)) (conjectured). (End)
%t A290194 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A290194 code = 705; stages = 128;
%t A290194 rule = IntegerDigits[code, 2, 10];
%t A290194 g = 2 * stages + 1; (* Maximum size of grid *)
%t A290194 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A290194 ca = a;
%t A290194 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A290194 PrependTo[ca, a];
%t A290194 (* Trim full grid to reflect growth by one cell at each stage *)
%t A290194 k = (Length[ca[[1]]] + 1)/2;
%t A290194 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 A290194 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A290194 Cf. A290192, A290193, A290195.
%K A290194 nonn,easy
%O A290194 0,2
%A A290194 _Robert Price_, Jul 23 2017