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A282802 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 507", based on the 5-celled von Neumann neighborhood. 4
1, 1, 7, 4, 31, 20, 127, 84, 511, 340, 2047, 1364, 8191, 5460, 32767, 21844, 131071, 87380, 524287, 349524, 2097151, 1398100, 8388607, 5592404, 33554431, 22369620, 134217727, 89478484, 536870911, 357913940, 2147483647, 1431655764, 8589934591, 5726623060 (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) = 2^(n+1) - 1 for n>1 and even.

a(n) = 2*(2^n - 2) / 3 for n>1 and odd.

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

G.f.: (1 + x + 2*x^2 - x^3 + 4*x^5) / ((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 = 507; 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. A282800, A282801, A282803.

Sequence in context: A282453 A282654 A282798 * A248278 A270235 A270719

Adjacent sequences:  A282799 A282800 A282801 * A282803 A282804 A282805

KEYWORD

nonn,easy

AUTHOR

Robert Price, Feb 21 2017

STATUS

approved

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Last modified December 15 09:05 EST 2019. Contains 329995 sequences. (Running on oeis4.)