A278661 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 57", based on the 5-celled von Neumann neighborhood.

1, 0, 1, 12, 5, 48, 7, 244, 25, 976, 7, 4084, 113, 16244, 21, 65472, 415, 261388, 97, 1048428, 1985, 4192028, 577, 16775196, 5057, 67093020, 6357, 268419616, 23183, 1073684512, 4015, 4294962736, 68279, 17179631792, 69175, 68719306412, 321389, 274876908426

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 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=57; 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. A278659, A278660, A278662.

Sequence in context: A110185 A299516 A278423 * A280614 A281041 A038331

Adjacent sequences: A278658 A278659 A278660 * A278662 A278663 A278664

Keyword

nonn,easy

Author

Robert Price, Nov 25 2016

Status

approved