A278345 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 35", based on the 5-celled von Neumann neighborhood.

1, 1, 1, 13, 1, 61, 1, 245, 5, 1013, 5, 4085, 5, 16341, 21, 65365, 85, 261973, 85, 1048405, 85, 4194133, 85, 16777045, 85, 67108181, 341, 268432725, 1365, 1073730901, 5461, 4294923605, 21845, 17179825493, 21845, 68719433045, 21845, 274877863253, 21845

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 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

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=35; 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. A278343, A278344, A278346.

Sequence in context: A046733 A357312 A291449 * A277866 A278594 A120392

Adjacent sequences: A278342 A278343 A278344 * A278346 A278347 A278348

Keyword

nonn,easy,changed

Author

Robert Price, Nov 18 2016

Status

approved