A277866 Decimal 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 3", based on the 5-celled von Neumann neighborhood.

1, 1, 1, 13, 1, 61, 1, 253, 1, 1021, 1, 4093, 1, 16381, 1, 65533, 1, 262141, 1, 1048573, 1, 4194301, 1, 16777213, 1, 67108861, 1, 268435453, 1, 1073741821, 1, 4294967293, 1, 17179869181, 1, 68719476733, 1, 274877906941, 1, 1099511627773, 1, 4398046511101, 1

Offset

0,4

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

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, Nov 03 2016: (Start)

G.f.: (1+x-4*x^2+8*x^3) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)).

a(n) = (-1-(-2)^n+2*(-1)^n+2^n). (End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=3; 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]], 2], {i, 1, stages-1}]

Crossrefs

Cf. A277864, A277865, A277867.

Sequence in context: A357312 A291449 A278345 * A278594 A120392 A133371

Adjacent sequences: A277863 A277864 A277865 * A277867 A277868 A277869

Keyword

nonn,easy

Author

Robert Price, Nov 02 2016

Status

approved