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A282653 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 491", based on the 5-celled von Neumann neighborhood. 4
1, 10, 111, 10, 11111, 10, 1111111, 100010, 111111111, 10000010, 11111111111, 1010100010, 1111111111111, 101010000010, 111111111111111, 10101010100010, 11111111111111111, 1010101010000010, 1111111111111111111, 101010101010100010, 111111111111111111111 (list; graph; refs; listen; history; text; internal format)
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, Feb 20 2017: (Start)

a(n) = 100*a(n-2) + a(n-4) - 100*a(n-6) for n>5.

G.f.: (1 + 10*x + 11*x^2 - 990*x^3 + 10*x^4 - 1000*x^5 + 100000*x^7 + 10000000*x^11) / ((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 = 491; 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. A282652, A282654, A282655.

Sequence in context: A282258 A100751 A282452 * A282801 A228006 A004290

Adjacent sequences:  A282650 A282651 A282652 * A282654 A282655 A282656

KEYWORD

nonn,easy

AUTHOR

Robert Price, Feb 20 2017

STATUS

approved

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Last modified March 20 15:43 EDT 2019. Contains 321345 sequences. (Running on oeis4.)