cp's OEIS Frontend

A287199 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 259", based on the 5-celled von Neumann neighborhood.

%I A287199 #13 Feb 16 2025 08:33:46
%S A287199 1,3,4,3,16,15,64,63,256,255,1024,1023,4096,4095,16384,16383,65536,
%T A287199 65535,262144,262143,1048576,1048575,4194304,4194303,16777216,
%U A287199 16777215,67108864,67108863,268435456,268435455,1073741824,1073741823,4294967296,4294967295
%N A287199 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 259", based on the 5-celled von Neumann neighborhood.
%C A287199 Initialized with a single black (ON) cell at stage zero.
%C A287199 Appears to differ from A277800 only at a(1). - _R. J. Mathar_, May 25 2017
%D A287199 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A287199 Robert Price, <a href="/A287199/b287199.txt">Table of n, a(n) for n = 0..126</a>
%H A287199 Robert Price, <a href="/A287199/a287199.tmp.txt">Diagrams of first 20 stages</a>
%H A287199 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 A287199 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A287199 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A287199 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A287199 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A287199 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A287199 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A287199 Conjectures from _Colin Barker_, May 25 2017: (Start)
%F A287199 G.f.: (1 + 3*x - x^2 - 12*x^3 + 12*x^5) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
%F A287199 a(n) = 2^n for n>1 and even.
%F A287199 a(n) = 2^(n-1) - 1 for n odd.
%F A287199 a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
%F A287199 (End)
%t A287199 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A287199 code = 259; stages = 128;
%t A287199 rule = IntegerDigits[code, 2, 10];
%t A287199 g = 2 * stages + 1; (* Maximum size of grid *)
%t A287199 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A287199 ca = a;
%t A287199 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A287199 PrependTo[ca, a];
%t A287199 (* Trim full grid to reflect growth by one cell at each stage *)
%t A287199 k = (Length[ca[[1]]] + 1)/2;
%t A287199 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 A287199 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A287199 Cf. A287194, A287196, A287197.
%K A287199 nonn,easy
%O A287199 0,2
%A A287199 _Robert Price_, May 21 2017