A279878 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 213", based on the 5-celled von Neumann neighborhood.

1, 1, 10, 11, 11100, 111, 1111000, 1111, 111110000, 11111, 11111100000, 111111, 1111111000000, 1111111, 111111110000000, 11111111, 11111111100000000, 111111111, 1111111111000000000, 1111111111, 111111111110000000000, 11111111111, 11111111111100000000000

Offset

0,3

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 21 2016: (Start)

a(n) = 111*a(n-2)-1110*a(n-4)+1000*a(n-6) for n>6.

G.f.: (1 +x -101*x^2 -100*x^3 +11100*x^4 -111000*x^6 +100000*x^8) / ((1 -x)*(1 +x)*(1 -10*x)*(1 +10*x)*(1 -10*x^2)).

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 213; 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. A279877, A279879, A279880.

Sequence in context: A279670 A279178 A279469 * A279138 A279753 A279146

Adjacent sequences: A279875 A279876 A279877 * A279879 A279880 A279881

Keyword

nonn,easy

Author

Robert Price, Dec 21 2016

Status

approved