A273972 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 20", based on the 5-celled von Neumann neighborhood.

1, 2, 4, 8, 16, 34, 68, 128, 256, 544, 1088, 2056, 4112, 8738, 17476, 32768, 65536, 139264, 278528, 526336, 1052672, 2236928, 4473856, 8388736, 16777472, 35652128, 71304256, 134744072, 269488144, 572662306, 1145324612, 2147483648, 4294967296, 9126805504

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 the 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

a(2*n+2) = 2*a(2*n+1). - Paul Curtz, Nov 16 2016

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=20; 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. A273495, A273531, A273973.

Sequence in context: A161869 A210541 A275545 * A275443 A288170 A088325

Adjacent sequences: A273969 A273970 A273971 * A273973 A273974 A273975

Keyword

nonn,easy

Author

Robert Price, Nov 09 2016

Status

approved