cp's OEIS Frontend

A290074 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 641", based on the 5-celled von Neumann neighborhood.

%I A290074 #14 Feb 16 2025 08:33:49
%S A290074 1,1,5,3,23,15,95,63,383,255,1535,1023,6143,4095,24575,16383,98303,
%T A290074 65535,393215,262143,1572863,1048575,6291455,4194303,25165823,
%U A290074 16777215,100663295,67108863,402653183,268435455,1610612735,1073741823,6442450943,4294967295
%N A290074 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 641", based on the 5-celled von Neumann neighborhood.
%C A290074 Initialized with a single black (ON) cell at stage zero.
%D A290074 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A290074 Robert Price, <a href="/A290074/b290074.txt">Table of n, a(n) for n = 0..126</a>
%H A290074 Robert Price, <a href="/A290074/a290074.tmp.txt">Diagrams of first 20 stages</a>
%H A290074 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 A290074 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A290074 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A290074 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A290074 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A290074 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A290074 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A290074 Conjectures from _Colin Barker_, Jul 20 2017: (Start)
%F A290074 G.f.: (1 - 2*x^3 + 4*x^4) / ((1 - x)*(1 - 2*x)*(1 + 2*x)).
%F A290074 a(n) = 3*2^(n-1) - 1 = A283507(n) for n>1 and even.
%F A290074 a(n) = 2^(n-1) - 1 = A283507(n) for n>1 and odd.
%F A290074 a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n>4.
%F A290074 (End)
%t A290074 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A290074 code = 641; stages = 128;
%t A290074 rule = IntegerDigits[code, 2, 10];
%t A290074 g = 2 * stages + 1; (* Maximum size of grid *)
%t A290074 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A290074 ca = a;
%t A290074 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A290074 PrependTo[ca, a];
%t A290074 (* Trim full grid to reflect growth by one cell at each stage *)
%t A290074 k = (Length[ca[[1]]] + 1)/2;
%t A290074 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 A290074 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A290074 Cf. A290070, A290072, A290073.
%K A290074 nonn,easy
%O A290074 0,3
%A A290074 _Robert Price_, Jul 19 2017