A280368 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 259", based on the 5-celled von Neumann neighborhood.

1, 10, 110, 1, 11000, 111, 1100000, 11111, 110000000, 1111111, 11000000000, 111111111, 1100000000000, 11111111111, 110000000000000, 1111111111111, 11000000000000000, 111111111111111, 1100000000000000000, 11111111111111111, 110000000000000000000

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 Colin Barker, Jan 01 2017: (Start)

a(n) = (989*(-10)^n + 991*10^n)/1800 for n>1 and even.

a(n) = (989*(-10)^n + 991*10^n - 200)/1800 for n>1 and odd.

a(n) = 101*a(n-2) - 100*a(n-4) for n>5.

G.f.: (1 + 10*x + 9*x^2 - 1009*x^3 - 10*x^4 + 1010*x^5) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 259; 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. A280367, A280369, A280370.

Sequence in context: A280834 A281305 A281415 * A280978 A281044 A281519

Adjacent sequences: A280365 A280366 A280367 * A280369 A280370 A280371

Keyword

nonn,easy

Author

Robert Price, Jan 01 2017

Status

approved