A286772 Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 221", based on the 5-celled von Neumann neighborhood.
1, 2, 0, 14, 1, 62, 1, 254, 1, 1022, 1, 4094, 1, 16382, 1, 65534, 1, 262142, 1, 1048574, 1, 4194302, 1, 16777214, 1, 67108862, 1, 268435454, 1, 1073741822, 1, 4294967294, 1, 17179869182, 1, 68719476734, 1, 274877906942, 1, 1099511627774, 1, 4398046511102, 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
- 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
- Wolfram Research, Wolfram Atlas of Simple Programs
- 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 = 221; 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[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
Formula
Conjectures from Colin Barker, May 14 2017: (Start)
G.f.: (1 + 2*x - 5*x^2 + 4*x^3 + 5*x^4 - 4*x^6) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
a(n) = 1 for n>2.
a(n) = 2^(n+1) - 2 for n>2.
a(n) = 5*a(n-2) - 4*a(n-4) for n>4.
(End)
It appears that a(n) = A280412(n) for n >= 3. - Michel Marcus, May 20 2017
Comments