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A279053 Binary 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 131", based on the 5-celled von Neumann neighborhood. 5
1, 1, 101, 101, 10101, 10101, 1010101, 1010101, 101010101, 101010101, 10101010101, 10101010101, 1010101010101, 1010101010101, 101010101010101, 101010101010101, 10101010101010101, 10101010101010101, 1010101010101010101, 1010101010101010101 (list; graph; refs; listen; history; text; internal format)
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 05 2016: (Start)

a(n) = (9*(-2)^n*5^(n+1) + 11*2^n*5^(n+1) - 1)/99.

a(n) = a(n-1) + 100*a(n-2) - 100*a(n-3) for n>2.

G.f.: 1 / ((1 - x)*(1 - 10*x)*(1 + 10*x)).

(End)

MATHEMATICA

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

code = 131; 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]], 10], {i, 1, stages - 1}]

CROSSREFS

Sequence in context: A283223 A279808 A279748 * A280145 A279528 A283044

Adjacent sequences:  A279050 A279051 A279052 * A279054 A279055 A279056

KEYWORD

nonn,easy

AUTHOR

Robert Price, Dec 04 2016

STATUS

approved

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Last modified December 15 14:37 EST 2019. Contains 329999 sequences. (Running on oeis4.)