cp's OEIS Frontend

A277799 Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 1", based on the 5-celled von Neumann neighborhood.

%I A277799 #18 Feb 16 2025 08:33:37
%S A277799 1,0,1,12,1,60,1,252,1,1020,1,4092,1,16380,1,65532,1,262140,1,1048572,
%T A277799 1,4194300,1,16777212,1,67108860,1,268435452,1,1073741820,1,
%U A277799 4294967292,1,17179869180,1,68719476732,1,274877906940,1,1099511627772,1,4398046511100,1
%N A277799 Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 1", based on the 5-celled von Neumann neighborhood.
%C A277799 Initialized with a single black (ON) cell at stage zero.
%C A277799 Rule numbers 1, 9, 17, 25, 257, 265, 273 and 281 all generate this sequence.
%D A277799 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A277799 Robert Price, <a href="/A277799/b277799.txt">Table of n, a(n) for n = 0..126</a>
%H A277799 Mattia Fregola, <a href="https://docs.google.com/spreadsheets/d/1629UXZ07lVK1-LVR0T7u1RDVKeW4f55K688CAtS5maw/edit?usp=sharing">Elementary Cellular Automata Rule 1 generating OEIS sequence A277799, A058896, A141725, A002450</a>
%H A277799 Robert Price, <a href="/A277799/a277799.tmp.txt">Diagrams of first 20 stages</a>
%H A277799 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 A277799 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A277799 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A277799 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A277799 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A277799 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A277799 Conjectures from _Colin Barker_, Nov 01 2016: (Start)
%F A277799 G.f.: (1 - 4*x^2 + 12*x^3) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
%F A277799 a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
%F A277799 a(n) = (-3/2-(-2)^n+(5*(-1)^n)/2+2^n). (End)
%t A277799 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A277799 code=1; stages=128;
%t A277799 rule=IntegerDigits[code,2,10];
%t A277799 g=2*stages+1; (* Maximum size of grid *)
%t A277799 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A277799 ca=a;
%t A277799 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A277799 PrependTo[ca,a];
%t A277799 (* Trim full grid to reflect growth by one cell at each stage *)
%t A277799 k=(Length[ca[[1]]]+1)/2;
%t A277799 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 A277799 Table[FromDigits[Part[ca[[i]][[i]],Range[1,i]],2], {i,1,stages-1}]
%Y A277799 Cf. A277797, A277798, A277800.
%K A277799 nonn,easy
%O A277799 0,4
%A A277799 _Robert Price_, Oct 31 2016