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A284177 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 809", based on the 5-celled von Neumann neighborhood. 4
1, 0, 1, 0, 5, 0, 21, 8, 73, 8, 329, 136, 1161, 168, 5289, 2216, 18601, 2728, 84137, 35496, 297129, 43688, 1353897, 567976, 4761769, 699048, 21538985, 9087656, 76064937, 11184808, 346597545, 145402536, 1219012777, 178956968, 5513980073, 2326440616 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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 + 16*x^6 + 8*x^7 - 204*x^8 + 256*x^10 + 128*x^11 - 192*x^12 + 32*x^13 + 32*x^14 + 512*x^17) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + 4*x^2)*(1 + 16*x^4)).

a(n) = a(n-2) + 256*a(n-8) - 256*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[1, i]], 2], {i, 1, stages - 1}]

CROSSREFS

Cf. A284175, A284176, A284178.

Sequence in context: A281742 A283011 A284135 * A222755 A279500 A185246

Adjacent sequences:  A284174 A284175 A284176 * A284178 A284179 A284180

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 21 2017

STATUS

approved

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Last modified November 13 23:01 EST 2019. Contains 329106 sequences. (Running on oeis4.)