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A277797 Binary 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 1", based on the 5-celled von Neumann neighborhood. 4
1, 0, 1, 1100, 1, 111100, 1, 11111100, 1, 1111111100, 1, 111111111100, 1, 11111111111100, 1, 1111111111111100, 1, 111111111111111100, 1, 11111111111111111100, 1, 1111111111111111111100, 1, 111111111111111111111100, 1, 11111111111111111111111100, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Initialized with a single black (ON) cell at stage zero.

Rule numbers 1, 9, 17, 25, 257, 265, 273 and 281 all generate this sequence.

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

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, Nov 01 2016: (Start)

G.f.: (1 - 100*x^2 + 1100*x^3)/((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).

a(n) = 101*a(n-2) - 100*a(n-4) for n>3.

a(n) = (-91+109*(-1)^n+10^(1+n)-(-1)^n*10^(1+n))/18. (End)

MATHEMATICA

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

code=1; 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]], 10], {i, 1, stages-1}]

CROSSREFS

Cf. A277798, A277799, A277800.

Sequence in context: A252418 A261911 A276708 * A278466 A280973 A260592

Adjacent sequences:  A277794 A277795 A277796 * A277798 A277799 A277800

KEYWORD

nonn,easy

AUTHOR

Robert Price, Oct 31 2016

STATUS

approved

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Last modified April 7 15:56 EDT 2020. Contains 333306 sequences. (Running on oeis4.)