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A280369 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 259", based on the 5-celled von Neumann neighborhood. 4
1, 1, 3, 8, 3, 56, 3, 248, 3, 1016, 3, 4088, 3, 16376, 3, 65528, 3, 262136, 3, 1048568, 3, 4194296, 3, 16777208, 3, 67108856, 3, 268435448, 3, 1073741816, 3, 4294967288, 3, 17179869176, 3, 68719476728, 3, 274877906936, 3, 1099511627768, 3, 4398046511096, 3 (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, Jan 01 2017: (Start)

a(n) = 3 for n>1 and even.

a(n) = 2^(n+1) - 8 for n>1 and odd.

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

G.f.: (1 + x - 2*x^2 + 3*x^3 - 8*x^4 + 20*x^5) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).

(End)

MATHEMATICA

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

code = 259; 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. A280367, A280368, A280370.

Sequence in context: A046970 A322360 A058936 * A280979 A281045 A002017

Adjacent sequences:  A280366 A280367 A280368 * A280370 A280371 A280372

KEYWORD

nonn,easy

AUTHOR

Robert Price, Jan 01 2017

STATUS

approved

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Last modified April 23 06:08 EDT 2019. Contains 322381 sequences. (Running on oeis4.)