A290193 Binary representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.

1, 1, 101, 1011, 111, 101111, 11111, 10111111, 1111111, 1011111111, 111111111, 101111111111, 11111111111, 10111111111111, 1111111111111, 1011111111111111, 111111111111111, 101111111111111111, 11111111111111111, 10111111111111111111, 1111111111111111111

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

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

Conjecture: For even n > 3, binary representation of 3 * 2^(n - 1) - 1. For odd n > 3, binary representation of 2^(n - 1) - 1. - David A. Corneth, Jul 23 2017

From Chai Wah Wu, Nov 01 2018: (Start)

a(n) = a(n-1) + 100*a(n-2) - 100*a(n-3) for n > 5 (conjectured).

G.f.: (10000*x^5 - 10900*x^4 + 910*x^3 + 1)/((x - 1)*(10*x - 1)*(10*x + 1)) (conjectured). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 705; 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[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

Crossrefs

Cf. A290192, A290194, A290195.

Sequence in context: A115775 A115800 A267041 * A289462 A289578 A190619

Adjacent sequences: A290190 A290191 A290192 * A290194 A290195 A290196

Keyword

nonn,easy

Author

Robert Price, Jul 23 2017

Status

approved