cp's OEIS Frontend

A277928 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 5", based on the 5-celled von Neumann neighborhood.

%I A277928 #10 Feb 16 2025 08:33:37
%S A277928 1,2,0,15,0,63,0,255,0,1023,0,4095,0,16383,0,65535,0,262143,0,1048575,
%T A277928 0,4194303,0,16777215,0,67108863,0,268435455,0,1073741823,0,
%U A277928 4294967295,0,17179869183,0,68719476735,0,274877906943,0,1099511627775,0,4398046511103,0
%N A277928 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 5", based on the 5-celled von Neumann neighborhood.
%C A277928 Initialized with a single black (ON) cell at stage zero.
%D A277928 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A277928 Robert Price, <a href="/A277928/b277928.txt">Table of n, a(n) for n = 0..126</a>
%H A277928 Robert Price, <a href="/A277928/a277928.tmp.txt">Diagrams of first 20 stages</a>
%H A277928 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 A277928 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A277928 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A277928 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A277928 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A277928 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A277928 Robert Price, <a href="/A277928/a277928.tmp.txt">Diagrams of first 20 stages</a>
%F A277928 Conjectures from _Colin Barker_, Nov 05 2016: (Start)
%F A277928 a(n) = 0 for n>1 and even; a(n) = 2^n-1-(-2)^n for n>1 and odd.
%F A277928 a(n) = 5*a(n-2)-4*a(n-4) for n>5.
%F A277928 G.f.: (1+2*x-5*x^2+5*x^3+4*x^4-4*x^5) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)).
%F A277928 (End)
%t A277928 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A277928 code=5; stages=128;
%t A277928 rule=IntegerDigits[code,2,10];
%t A277928 g=2*stages+1; (* Maximum size of grid *)
%t A277928 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A277928 ca=a;
%t A277928 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A277928 PrependTo[ca,a];
%t A277928 (* Trim full grid to reflect growth by one cell at each stage *)
%t A277928 k=(Length[ca[[1]]]+1)/2;
%t A277928 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 A277928 Table[FromDigits[Part[ca[[i]][[i]],Range[1,i]],2], {i,1,stages-1}]
%Y A277928 Cf. A277926, A277927, A277929.
%K A277928 nonn,easy
%O A277928 0,2
%A A277928 _Robert Price_, Nov 04 2016