A285473 Binary 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 3", based on the 5-celled von Neumann neighborhood.
1, 11, 1, 1111, 1, 111111, 1, 11111111, 1, 1111111111, 1, 111111111111, 1, 11111111111111, 1, 1111111111111111, 1, 111111111111111111, 1, 11111111111111111111, 1, 1111111111111111111111, 1, 111111111111111111111111, 1, 11111111111111111111111111, 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
- Robert Price, Diagrams of first 20 stages
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 = 3; 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, Apr 19 2017: (Start)
G.f.: (1 + 11*x - 100*x^2) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
a(n) = (4 + 5*(-1)^n - (-2)^n*5^(1+n) + 2^n*5^(1+n))/9.
a(n) = 101*a(n-2) - 100*a(n-4) for n>3.
(End)
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