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A278446 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 43", based on the 5-celled von Neumann neighborhood. 4
1, 2, 4, 11, 20, 43, 80, 175, 336, 687, 1344, 2751, 5440, 10943, 21760, 43775, 87296, 174847, 349184, 699391, 1397760, 2796543, 5591040, 11186175, 22368256, 44740607, 89473024, 178962431, 357908480, 715833343, 1431633920, 2863333375, 5726601216, 11453267967 (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

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

FORMULA

Empirical g.f.: (1 + 2*x) * (1 - 2*x + 3*x^2 - 3*x^3) / ((1 - x) * (1 + x) * (1 - 2*x) * (1 - 2*x^2) * (1 + 2*x^2)). - Colin Barker, Nov 23 2016

MATHEMATICA

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

code=43; 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. A278443, A278444, A278445.

Sequence in context: A204519 A076636 A011954 * A026275 A152597 A320679

Adjacent sequences:  A278443 A278444 A278445 * A278447 A278448 A278449

KEYWORD

nonn,easy

AUTHOR

Robert Price, Nov 22 2016

STATUS

approved

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Last modified October 23 04:06 EDT 2021. Contains 348211 sequences. (Running on oeis4.)