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A285613 Binary representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 62", based on the 5-celled von Neumann neighborhood. 2
1, 10, 10, 100, 1100, 11000, 111000, 1110000, 11110000, 111100000, 1111100000, 11111000000, 111111000000, 1111110000000, 11111110000000, 111111100000000, 1111111100000000, 11111111000000000, 111111111000000000, 1111111110000000000, 11111111110000000000 (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, Apr 23 2017: (Start)

G.f.: (1 - 100*x^2 + 1000*x^4) / ((1 - 10*x)*(1 - 10*x^2)).

a(n) = (10^n - 10^(n/2)) / 9 for n>1 and even.

a(n) = (10^n - 10^(n/2+1/2)) / 9 for n>1 and odd.

a(n) = 10*a(n-1) + 10*a(n-2) - 100*a(n-3) for n>2.

(End)

MATHEMATICA

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

code = 62; 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. A285612, A056453, A233411.

Sequence in context: A285942 A286408 A285609 * A288301 A287602 A165428

Adjacent sequences:  A285610 A285611 A285612 * A285614 A285615 A285616

KEYWORD

nonn,easy

AUTHOR

Robert Price, Apr 22 2017

STATUS

approved

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Last modified November 29 23:37 EST 2020. Contains 338780 sequences. (Running on oeis4.)