A290662 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 899", based on the 5-celled von Neumann neighborhood.
1, 3, 5, 15, 23, 63, 95, 255, 383, 1023, 1535, 4095, 6143, 16383, 24575, 65535, 98303, 262143, 393215, 1048575, 1572863, 4194303, 6291455, 16777215, 25165823, 67108863, 100663295, 268435455, 402653183, 1073741823, 1610612735, 4294967295, 6442450943
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 = 899; 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 Chai Wah Wu, Aug 03 2020: (Start)
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n > 3.
G.f.: (2*x^3 - 2*x^2 + 2*x + 1)/((x - 1)*(2*x - 1)*(2*x + 1)). (End)
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