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A284178 Decimal 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 809", based on the 5-celled von Neumann neighborhood. 4
1, 0, 4, 0, 20, 0, 84, 16, 292, 64, 1172, 272, 4644, 1344, 19092, 5392, 76324, 21824, 305300, 87312, 1221156, 349504, 4885140, 1398032, 19540516, 5592384, 78162068, 22369552, 312648228, 89478464, 1250593428, 357913872, 5002373668, 1431655744, 20009494676 (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, Mar 22 2017: (Start)

G.f.: (1 + 4*x^4 + 4*x^6 + 16*x^7 - 45*x^8 + 4*x^10 + 16*x^11 - 48*x^12 + 256*x^13 + 512*x^14 + 256*x^17) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + x^2)*(1 + x^4)).

a(n) = 4*a(n-2) + a(n-8) - 4*a(n-10) for n>14.

(End)

MATHEMATICA

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

code = 809; 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]], 2], {i , 1, stages - 1}]

CROSSREFS

Cf. A284175, A284176, A284177.

Sequence in context: A282579 A283012 A284136 * A286032 A199933 A078630

Adjacent sequences:  A284175 A284176 A284177 * A284179 A284180 A284181

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 21 2017

STATUS

approved

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Last modified August 9 01:35 EDT 2020. Contains 336310 sequences. (Running on oeis4.)