A278756 Decimal 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 65", based on the 5-celled von Neumann neighborhood.

1, 0, 6, 1, 28, 3, 120, 7, 496, 15, 2016, 31, 8128, 63, 32640, 127, 130816, 255, 523776, 511, 2096128, 1023, 8386560, 2047, 33550336, 4095, 134209536, 8191, 536854528, 16383, 2147450880, 32767, 8589869056, 65535, 34359607296, 131071, 137438691328, 262143

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

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

Robert Price, Diagrams of first 20 stages

Formula

Conjectures from Chai Wah Wu, Jun 15 2020: (Start)

a(n) = 7*a(n-2) - 14*a(n-4) + 8*a(n-6) for n > 5.

G.f.: (4*x^5 - x^3 + x^2 - 1)/(8*x^6 - 14*x^4 + 7*x^2 - 1). (End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=65; 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]], 2], {i, 1, stages-1}]

Crossrefs

Cf. A278753, A278754, A278755.

Sequence in context: A278958 A281631 A259230 * A147327 A145629 A193633

Adjacent sequences: A278753 A278754 A278755 * A278757 A278758 A278759

Keyword

nonn,easy

Author

Robert Price, Nov 27 2016

Status

approved