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A282803 Decimal 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 507", based on the 5-celled von Neumann neighborhood. 4
1, 2, 7, 2, 31, 10, 127, 42, 511, 170, 2047, 682, 8191, 2730, 32767, 10922, 131071, 43690, 524287, 174762, 2097151, 699050, 8388607, 2796202, 33554431, 11184810, 134217727, 44739242, 536870911, 178956970, 2147483647, 715827882, 8589934591, 2863311530 (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

From Colin Barker, Feb 22 2017: (Start)

Conjectures:

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

a(n) = (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 + 2*x + 2*x^2 - 8*x^3 + 8*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[i, 2 * i - 1]], 2], {i , 1, stages - 1}]

CROSSREFS

Cf. A282800, A282801, A282802.

Sequence in context: A176379 A282454 A282655 * A248683 A110987 A199740

Adjacent sequences:  A282800 A282801 A282802 * A282804 A282805 A282806

KEYWORD

nonn,easy

AUTHOR

Robert Price, Feb 21 2017

STATUS

approved

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Last modified April 6 12:08 EDT 2020. Contains 333273 sequences. (Running on oeis4.)