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A282827 Binary 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 521", based on the 5-celled von Neumann neighborhood. 4
1, 0, 100, 0, 11100, 111000, 1010100, 0, 101111100, 11111000, 10101010100, 0, 1011011111100, 110101111000, 101100011010100, 11010110000000, 10110001101111100, 1100011011111000, 1011000110101010100, 110001100000000000, 101100011011011111100 (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, Feb 22 2017: (Start)

a(n) = 100*a(n-2) + a(n-8) - 100*a(n-10) for n>15.

G.f.: (1 + 1100*x^4 + 111000*x^5 - 99900*x^6 - 11100000*x^7 + 101099*x^8 + 11111000*x^9 - 10099900*x^10 - 1111100000*x^11 + 910100000*x^12 + 110101000000*x^13 - 1100000000*x^14 + 10000000*x^15 - 1000000000000*x^17) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)*(1 + x^2)*(1 + x^4)).

(End)

MATHEMATICA

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

code = 521; 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. A282826, A282828, A282829.

Sequence in context: A282977 A284084 A282805 * A283915 A284184 A283066

Adjacent sequences:  A282824 A282825 A282826 * A282828 A282829 A282830

KEYWORD

nonn,easy

AUTHOR

Robert Price, Feb 22 2017

STATUS

approved

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Last modified November 17 23:26 EST 2019. Contains 329242 sequences. (Running on oeis4.)