A279875 Decimal representation of the x-axis, from the left edge to the origin, 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, 1, 5, 7, 16, 23, 84, 125, 257, 383, 1344, 2015, 4096, 6143, 21504, 32255, 65536, 98303, 344064, 516095, 1048576, 1572863, 5505024, 8257535, 16777216, 25165823, 88080384, 132120575, 268435456, 402653183, 1409286144, 2113929215, 4294967296, 6442450943

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) = a(n-2) + 16*a(n-4) - 16*a(n-6) for n>10.

G.f.: (1 +x +4*x^2 +6*x^3 -5*x^4 +4*x^6 +6*x^7 -3*x^8 +2*x^9 -x^10 -16*x^12 +16*x^14) / ((1 -x)*(1 +x)*(1 -2*x)*(1 +2*x)*(1 +4*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[1, i]], 2], {i, 1, stages - 1}]

Crossrefs

Cf. A279873, A279874, A279876.

Sequence in context: A272856 A218623 A279175 * A185872 A186710 A276717

Adjacent sequences: A279872 A279873 A279874 * A279876 A279877 A279878

Keyword

nonn,easy

Author

Robert Price, Dec 21 2016

Status

approved