A282261 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 446", based on the 5-celled von Neumann neighborhood.

1, 11, 110, 1111, 11000, 111000, 1101010, 11111111, 110000010, 1110000011, 11010101010, 111111111111, 1100000010010, 11100001010011, 110101001110010, 1111111001010011, 11000001001110010, 111000001001110011, 1101010101001110010, 11111111111001110011

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

Conjectures from Chai Wah Wu, May 05 2024: (Start)

a(n) = a(n-2) + 100000000*a(n-8) - 100000000*a(n-10) for n > 17.

G.f.: (100000*x^17 - 100000000*x^15 + 1100000*x^14 - 10101100*x^13 - 10091000*x^12 + 1111100*x^11 + 101000*x^10 - 1111100*x^9 + 8899000*x^8 + 11000111*x^7 + 1090010*x^6 + 109889*x^5 + 10890*x^4 + 1100*x^3 + 109*x^2 + 11*x + 1)/(100000000*x^10 - 100000000*x^8 - 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 = 446; 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. A282262, A282263, A282264.

Sequence in context: A285821 A281756 A280463 * A282006 A280099 A282385

Adjacent sequences: A282258 A282259 A282260 * A282262 A282263 A282264

Keyword

nonn,easy

Author

Robert Price, Feb 10 2017

Status

approved