A288137 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 427", based on the 5-celled von Neumann neighborhood.

1, 3, 3, 14, 7, 63, 14, 251, 31, 1022, 59, 4079, 126, 16379, 239, 65470, 507, 262127, 958, 1048315, 2031, 4194238, 3835, 16776175, 8126, 67108603, 15343, 268431294, 32507, 1073740783, 61374, 4294950651, 130031, 17179865022, 245499, 68719410159, 520126

Offset

0,2

Comments

Initialized with a single black (ON) cell at stage zero.

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

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

Formula

G.f.: (1 + 3*x - x^2 + x^3 - 12*x^4 - 4*x^5 - 12*x^6) / ((1 - x)*(1 - 2*x)*(1 + 2*x)*(1 + x + x^2)*(1 - 2*x^2)*(1 + 2*x^2)) (conjectured). - Colin Barker, Jun 06 2017

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 427; 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}]

Crossrefs

Cf. A288131, A288136, A288138.

Sequence in context: A287780 A288062 A288590 * A287501 A287537 A186373

Adjacent sequences: A288134 A288135 A288136 * A288138 A288139 A288140

Keyword

nonn,easy

Author

Robert Price, Jun 05 2017

Status

approved