A280414 Binary 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 262", based on the 5-celled von Neumann neighborhood.

1, 11, 1, 111, 10001, 100111, 1000001, 11000111, 1000001, 111000111, 10000000001, 100000000111, 1000000000001, 10000000000111, 100000000000001, 1000000000000111, 10000000000000001, 100000000000000111, 1000000000000000001, 10000000000000000111

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 Chai Wah Wu, May 05 2024: (Start)

a(n) = a(n-2) for n > 31.

G.f.: (110000000000000000000000000000*x^31 + 10000000000000000000000000000*x^30 - 109000000000000000000000000000*x^29 - 9900000000000000000000000000*x^28 - 990000000000000000000000000*x^27 - 99000000000000000000000000*x^26 - 9900000000000000000000000*x^25 - 990000000000000000000000*x^24 - 99000000000000000000000*x^23 - 9900000000000000000000*x^22 - 990000000000000000000*x^21 - 99000000000000000000*x^20 - 9900000000000000000*x^19 - 990000000000000000*x^18 - 99000000000000000*x^17 - 9900000000000000*x^16 - 990000000000000*x^15 - 99000000000000*x^14 - 9900000000000*x^13 - 990000000000*x^12 - 99889000000*x^11 - 9999000000*x^10 - 100000000*x^9 - 10900000*x^7 - 990000*x^6 - 100000*x^5 - 10000*x^4 - 100*x^3 - 11*x - 1)/(x^2 - 1). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 262; 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]], 10], {i, 1, stages - 1}]

Crossrefs

Cf. A280413, A280415, A280416.

Sequence in context: A279937 A340566 A280607 * A281736 A281107 A281215

Adjacent sequences: A280411 A280412 A280413 * A280415 A280416 A280417

Keyword

nonn,easy

Author

Robert Price, Jan 02 2017

Status

approved