A280373 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 260", based on the 5-celled von Neumann neighborhood.

1, 2, 5, 8, 16, 40, 84, 130, 257, 640, 1344, 2080, 4096, 10240, 21504, 33280, 65536, 163840, 344064, 532480, 1048576, 2621440, 5505024, 8519680, 16777216, 41943040, 88080384, 136314880, 268435456, 671088640, 1409286144, 2181038080, 4294967296, 10737418240

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 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, Jan 02 2017: (Start)

a(n) = 16*a(n-4) for n>3.

G.f.: (1 + 2*x + 5*x^2 + 8*x^3 + 8*x^5 + 4*x^6 + 2*x^7 + x^8 - 16*x^12) / ((1 - 2*x)*(1 + 2*x)*(1 + 4*x^2)).

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 260; 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. A280371, A280372, A280374.

Sequence in context: A357239 A071085 A240951 * A324612 A243189 A055236

Adjacent sequences: A280370 A280371 A280372 * A280374 A280375 A280376

Keyword

nonn,easy

Author

Robert Price, Jan 01 2017

Status

approved