A278757 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 73", based on the 5-celled von Neumann neighborhood.

1, 0, 11, 1010, 0, 111111, 0, 11111111, 0, 1111111111, 0, 111111111111, 0, 11111111111111, 0, 1111111111111111, 0, 111111111111111111, 0, 11111111111111111111, 0, 1111111111111111111111, 0, 111111111111111111111111, 0, 11111111111111111111111111, 0

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

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

Formula

Conjectures from Chai Wah Wu, Jun 14 2020: (Start)

a(n) = 101*a(n-2) - 100*a(n-4) for n > 7.

G.f.: (-10100*x^7 + 1100*x^6 + 9101*x^5 - 1011*x^4 + 1010*x^3 - 90*x^2 + 1)/(100*x^4 - 101*x^2 + 1). (End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=73; 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. A278758, A278759, A278760.

Sequence in context: A004656 A143016 A152577 * A281172 A281218 A280527

Adjacent sequences: A278754 A278755 A278756 * A278758 A278759 A278760

Keyword

nonn,easy

Author

Robert Price, Nov 27 2016

Status

approved