A279874 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 211", based on the 5-celled von Neumann neighborhood.

1, 10, 101, 1110, 1, 111010, 10101, 10111110, 100000001, 1111111010, 10101, 111110111110, 1, 11111111111010, 10101, 1111111110111110, 1, 111111111111111010, 10101, 11111111111110111110, 1, 1111111111111111111010, 10101, 111111111111111110111110, 1

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

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

Formula

Conjectures from Colin Barker, Dec 22 2016: (Start)

a(n) = 100*a(n-2) + a(n-4) - 100*a(n-6) for n>10.

G.f.: (1 +10*x +x^2 +110*x^3 -10100*x^4 +10000*x^6 -990000*x^7 +99000000*x^8 +100000000*x^9 -10000000000*x^10 -100000000*x^12 +10000000000*x^14) / ((1 -x)*(1 +x)*(1 -10*x)*(1 +10*x)*(1 +x^2)).

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 211; 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. A279873, A279875, A279876.

Sequence in context: A290196 A333686 A279174 * A279251 A279123 A317947

Adjacent sequences: A279871 A279872 A279873 * A279875 A279876 A279877

Keyword

nonn,easy

Author

Robert Price, Dec 21 2016

Status

approved