A283604 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 670", based on the 5-celled von Neumann neighborhood.

1, 11, 111, 1111, 11101, 111111, 1110111, 11111111, 111011101, 1111111111, 11101111011, 111111111111, 1110111110011, 11111111111111, 111011111111011, 1111111111111111, 11101111111111111, 111111111111111111, 1110111111111111111, 11111111111111111111

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 06 2024: (Start)

a(n) = a(n-1) + 100*a(n-2) - 100*a(n-3) for n > 17.

G.f.: (-10000*x^17 + 10000*x^16 - 109900*x^15 + 109900*x^14 - 8900*x^13 + 8900*x^12 - 900*x^11 + 900*x^10 + 10*x^9 - 10*x^8 + 10*x^5 - 10*x^4 + 10*x + 1)/((x - 1)*(10*x - 1)*(10*x + 1)). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 670; 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. A283605, A283606, A283607.

Sequence in context: A284087 A284419 A283849 * A090103 A061851 A247552

Adjacent sequences: A283601 A283602 A283603 * A283605 A283606 A283607

Keyword

nonn,easy,changed

Author

Robert Price, Mar 11 2017

Status

approved