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.
1, 0, 1, 12, 1, 60, 1, 252, 1, 1020, 1, 4092, 1, 16380, 1, 65532, 1, 262140, 1, 1048572, 1, 4194300, 1, 16777212, 1, 67108860, 1, 268435452, 1, 1073741820, 1, 4294967292, 1, 17179869180, 1, 68719476732, 1, 274877906940, 1, 1099511627772, 1, 4398046511100, 1
Offset: 0
References
- S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
Links
- Robert Price, Table of n, a(n) for n = 0..126
- Mattia Fregola, Elementary Cellular Automata Rule 1 generating OEIS sequence A277799, A058896, A141725, A002450
- Robert Price, Diagrams of first 20 stages
- N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- S. Wolfram, A New Kind of Science
- Index entries for sequences related to cellular automata
- Index to 2D 5-Neighbor Cellular Automata
- Index to Elementary Cellular Automata
Programs
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Mathematica
CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}]; code=1; stages=128; rule=IntegerDigits[code,2,10]; g=2*stages+1; (* Maximum size of grid *) a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *) ca=a; ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}]; PrependTo[ca,a]; (* Trim full grid to reflect growth by one cell at each stage *) k=(Length[ca[[1]]]+1)/2; ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}]; Table[FromDigits[Part[ca[[i]][[i]],Range[1,i]],2], {i,1,stages-1}]
Formula
Conjectures from Colin Barker, Nov 01 2016: (Start)
G.f.: (1 - 4*x^2 + 12*x^3) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
a(n) = (-3/2-(-2)^n+(5*(-1)^n)/2+2^n). (End)
Comments