A278424 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 41", based on the 5-celled von Neumann neighborhood.

1, 0, 4, 3, 20, 3, 112, 47, 272, 143, 1088, 575, 4416, 2111, 22272, 2815, 114944, 47359, 283648, 132095, 1168384, 558079, 4485120, 2273279, 22024192, 757759, 130367488, 44056575, 274546688, 237518847, 1158086656, 552271871, 4785766400, 2822307839

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

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=41; 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. A278421, A278422, A278423.

Sequence in context: A178628 A345125 A167479 * A278662 A220861 A270797

Adjacent sequences: A278421 A278422 A278423 * A278425 A278426 A278427

Keyword

nonn,easy

Author

Robert Price, Nov 21 2016

Status

approved