A281735 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 390", based on the 5-celled von Neumann neighborhood.

1, 11, 100, 1110, 10001, 111001, 1000001, 11100011, 100000100, 1110001110, 10000000001, 111000000001, 1000000000001, 11100000000001, 100000000000001, 1110000000000001, 10000000000000001, 111000000000000001, 1000000000000000001, 11100000000000000001

Offset

0,2

Comments

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

Differs from A280413 first at a(23). - R. J. Mathar, Feb 02 2017

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 Chai Wah Wu, May 05 2024: (Start)

a(n) = 100*a(n-2) for n > 25.

G.f.: (1100*x^25 + 100*x^24 + 89*x^23 + 99*x^22 + 99*x^21 + 99*x^20 + 99*x^19 + 99*x^18 + 99*x^17 + 99*x^16 + 99*x^15 + 99*x^14 + 99*x^13 + 99*x^12 + 110999*x^11 + 9999*x^10 - 10*x^9 + 89*x^7 + 99*x^6 - x^5 - x^4 - 10*x^3 - 11*x - 1)/(100*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 = 390; 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. A281736, A281737, A281738.

Sequence in context: A278951 A280606 A280413 * A281106 A281308 A281214

Adjacent sequences: A281732 A281733 A281734 * A281736 A281737 A281738

Keyword

nonn,easy,changed

Author

Robert Price, Jan 28 2017

Status

approved