cp's OEIS Frontend

A286773 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 221", based on the 5-celled von Neumann neighborhood.

%I A286773 #11 Feb 16 2025 08:33:45
%S A286773 1,1,0,7,16,31,64,127,256,511,1024,2047,4096,8191,16384,32767,65536,
%T A286773 131071,262144,524287,1048576,2097151,4194304,8388607,16777216,
%U A286773 33554431,67108864,134217727,268435456,536870911,1073741824,2147483647,4294967296,8589934591
%N A286773 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 221", based on the 5-celled von Neumann neighborhood.
%C A286773 Initialized with a single black (ON) cell at stage zero.
%D A286773 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A286773 Robert Price, <a href="/A286773/b286773.txt">Table of n, a(n) for n = 0..126</a>
%H A286773 Robert Price, <a href="/A286773/a286773.tmp.txt">Diagrams of first 20 stages</a>
%H A286773 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 A286773 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A286773 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A286773 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A286773 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A286773 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A286773 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A286773 Conjectures from _Colin Barker_, May 15 2017: (Start)
%F A286773 G.f.: (1 - x - 3*x^2 + 8*x^3 + 4*x^4 - 8*x^5) / ((1 - x)*(1 + x)*(1 - 2*x)).
%F A286773 a(n) = 2^n for n>2 and even.
%F A286773 a(n) = 2^n - 1 for n>2 and odd.
%F A286773 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>5.
%F A286773 (End)
%t A286773 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A286773 code = 221; stages = 128;
%t A286773 rule = IntegerDigits[code, 2, 10];
%t A286773 g = 2 * stages + 1; (* Maximum size of grid *)
%t A286773 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A286773 ca = a;
%t A286773 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A286773 PrependTo[ca, a];
%t A286773 (* Trim full grid to reflect growth by one cell at each stage *)
%t A286773 k = (Length[ca[[1]]] + 1)/2;
%t A286773 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 A286773 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A286773 Cf. A286770, A286771, A286772.
%K A286773 nonn,easy
%O A286773 0,4
%A A286773 _Robert Price_, May 14 2017