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A286772 Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 221", based on the 5-celled von Neumann neighborhood. 4
1, 2, 0, 14, 1, 62, 1, 254, 1, 1022, 1, 4094, 1, 16382, 1, 65534, 1, 262142, 1, 1048574, 1, 4194302, 1, 16777214, 1, 67108862, 1, 268435454, 1, 1073741822, 1, 4294967294, 1, 17179869182, 1, 68719476734, 1, 274877906942, 1, 1099511627774, 1, 4398046511102, 1 (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, May 14 2017: (Start)

G.f.: (1 + 2*x - 5*x^2 + 4*x^3 + 5*x^4 - 4*x^6) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).

a(n) = 1 for n>2.

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

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

(End)

It appears that a(n) = A280412(n) for n >= 3. - Michel Marcus, May 20 2017

MATHEMATICA

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

code = 221; 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. A286770, A286771, A286773.

Sequence in context: A286118 A286084 A286732 * A286505 A286405 A287131

Adjacent sequences:  A286769 A286770 A286771 * A286773 A286774 A286775

KEYWORD

nonn,easy

AUTHOR

Robert Price, May 14 2017

STATUS

approved

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Last modified June 19 05:33 EDT 2019. Contains 324218 sequences. (Running on oeis4.)