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A282123 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 430", based on the 5-celled von Neumann neighborhood. 4
1, 3, 6, 15, 26, 63, 106, 255, 426, 1023, 1706, 4095, 6826, 16383, 27306, 65535, 109226, 262143, 436906, 1048575, 1747626, 4194303, 6990506, 16777215, 27962026, 67108863, 111848106, 268435455, 447392426, 1073741823, 1789569706, 4294967295, 7158278826 (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, Feb 07 2017: (Start)

a(n) = (-5 - (-2)^n + (-1)^n + 11*2^n) / 6.

a(n) = 5*a(n-2) - 4*a(n-4) for n>3.

G.f.: (1 + 3*x + x^2) / ((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 = 430; 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. A282121, A282122, A282124.

Sequence in context: A327067 A252745 A282662 * A281420 A280839 A134774

Adjacent sequences:  A282120 A282121 A282122 * A282124 A282125 A282126

KEYWORD

nonn,easy

AUTHOR

Robert Price, Feb 06 2017

STATUS

approved

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Last modified July 8 01:42 EDT 2020. Contains 335502 sequences. (Running on oeis4.)