A277798 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 1", based on the 5-celled von Neumann neighborhood.

1, 0, 100, 11, 10000, 1111, 1000000, 111111, 100000000, 11111111, 10000000000, 1111111111, 1000000000000, 111111111111, 100000000000000, 11111111111111, 10000000000000000, 1111111111111111, 1000000000000000000, 111111111111111111, 100000000000000000000

Offset

0,3

Comments

Initialized with a single black (ON) cell at stage zero.

Rule numbers 1, 9, 17, 25, 257, 265, 273 and 281 all generate this sequence.

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 01 2016: (Start)

G.f.: (1 - x^2 + 11*x^3)/((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).

a(n) = 101*a(n-2) - 100*a(n-4) for n>3.

a(n) = (-10+89*(-10)^n+10*(-1)^n+91*10^n)/180. (End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=1; 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. A277797, A277799, A277800.

Sequence in context: A231891 A084472 A276768 * A278467 A280974 A285644

Adjacent sequences: A277795 A277796 A277797 * A277799 A277800 A277801

Keyword

nonn,easy

Author

Robert Price, Oct 31 2016

Status

approved