A278443 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 43", based on the 5-celled von Neumann neighborhood.

1, 1, 1, 1101, 101, 110101, 101, 11110101, 10101, 1111010101, 10101, 111111010101, 1010101, 11111101010101, 1010101, 1111111101010101, 101010101, 111111110101010101, 101010101, 11111111110101010101, 10101010101, 1111111111010101010101, 10101010101

Offset

0,4

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

Empirical g.f.: (1 - 100*x^2 + 1100*x^3 - 1100*x^4) / ((1 - x) * (1 - 10*x) * (1 + 10*x) * (1 - 10*x^2) * (1 + 10*x^2)). - Colin Barker, Nov 23 2016

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=43; 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. A278444, A278445, A278446.

Sequence in context: A278343 A277864 A278592 * A308208 A290684 A290852

Adjacent sequences: A278440 A278441 A278442 * A278444 A278445 A278446

Keyword

nonn,easy

Author

Robert Price, Nov 22 2016

Status

approved