|
| |
|
|
A279880
|
|
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 213", based on the 5-celled von Neumann neighborhood.
|
|
4
|
|
|
|
1, 1, 2, 3, 28, 7, 120, 15, 496, 31, 2016, 63, 8128, 127, 32640, 255, 130816, 511, 523776, 1023, 2096128, 2047, 8386560, 4095, 33550336, 8191, 134209536, 16383, 536854528, 32767, 2147450880, 65535, 8589869056, 131071, 34359607296, 262143, 137438691328
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
|
|
|
FORMULA
|
Conjectures from Colin Barker, Dec 22 2016: (Start)
a(n) = 7*a(n-2)-14*a(n-4)+8*a(n-6) for n>6.
G.f.: (1+x-5*x^2-4*x^3+28*x^4-56*x^6+32*x^8) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-2*x^2)).
(End)
|
|
|
MATHEMATICA
|
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 213; 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. A279877, A279878, A279879.
Sequence in context: A098812 A073049 A349826 * A279140 A279755 A279148
Adjacent sequences: A279877 A279878 A279879 * A279881 A279882 A279883
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
Robert Price, Dec 21 2016
|
|
|
STATUS
|
approved
|
| |
|
|
