cp's OEIS Frontend

A286521 Decimal representation of the diagonal from the origin to the corner (or of the corner to the origin) of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 659", based on the 5-celled von Neumann neighborhood.

%I A286521 #16 Feb 16 2025 08:33:45
%S A286521 1,3,5,15,31,63,127,255,511,1023,2047,4095,8191,16383,32767,65535,
%T A286521 131071,262143,524287,1048575,2097151,4194303,8388607,16777215,
%U A286521 33554431,67108863,134217727,268435455,536870911,1073741823,2147483647,4294967295,8589934591
%N A286521 Decimal representation of the diagonal from the origin to the corner (or of the corner to the origin) of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 659", based on the 5-celled von Neumann neighborhood.
%C A286521 Initialized with a single black (ON) cell at stage zero.
%D A286521 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A286521 Robert Price, <a href="/A286521/b286521.txt">Table of n, a(n) for n = 0..126</a>
%H A286521 Robert Price, <a href="/A286521/a286521.tmp.txt">Diagrams of first 20 stages</a>
%H A286521 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 A286521 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A286521 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A286521 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A286521 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A286521 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A286521 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A286521 Conjectures from _Colin Barker_, Jul 22 2017: (Start)
%F A286521 G.f.: (1 - 2*x^2 + 6*x^3 - 4*x^4) / ((1 - x)*(1 - 2*x)).
%F A286521 a(n) = 2^(1+n) - 1 for n>2.
%F A286521 a(n) = 3*a(n-1) - 2*a(n-2) for n>4.
%F A286521 (End)
%t A286521 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A286521 code = 659; stages = 128;
%t A286521 rule = IntegerDigits[code, 2, 10];
%t A286521 g = 2 * stages + 1; (* Maximum size of grid *)
%t A286521 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A286521 ca = a;
%t A286521 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A286521 PrependTo[ca, a];
%t A286521 (* Trim full grid to reflect growth by one cell at each stage *)
%t A286521 k = (Length[ca[[1]]] + 1)/2;
%t A286521 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 A286521 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A286521 Cf. A286518, A286519, A286520.
%K A286521 nonn,easy
%O A286521 0,2
%A A286521 _Robert Price_, Jul 22 2017