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A283506 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 641", based on the 5-celled von Neumann neighborhood. 5
1, 0, 5, 12, 29, 60, 125, 252, 509, 1020, 2045, 4092, 8189, 16380, 32765, 65532, 131069, 262140, 524285, 1048572, 2097149, 4194300, 8388605, 16777212, 33554429, 67108860, 134217725, 268435452, 536870909, 1073741820, 2147483645, 4294967292, 8589934589 (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, Mar 10 2017: (Start)

G.f.: (1 - 2*x + 4*x^2 + 4*x^3) / ((1 - x)*(1 + x)*(1 - 2*x)).

a(n) = 2^(n+1) - 3 for n>0 and even.

a(n) = 2^(n+1) - 4 for n odd.

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

(End)

MATHEMATICA

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

code = 641; 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. A283504, A283505, A283507.

Sequence in context: A055245 A196410 A000465 * A266471 A069306 A277088

Adjacent sequences:  A283503 A283504 A283505 * A283507 A283508 A283509

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 09 2017

STATUS

approved

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Last modified October 23 22:04 EDT 2018. Contains 316541 sequences. (Running on oeis4.)