A280415 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 262", based on the 5-celled von Neumann neighborhood.

1, 3, 4, 14, 17, 57, 65, 227, 260, 910, 1025, 3585, 4097, 14337, 16385, 57345, 65537, 229377, 262145, 917505, 1048577, 3670017, 4194305, 14680065, 16777217, 58720257, 67108865, 234881025, 268435457, 939524099, 1073741824, 3758096384, 4294967296, 15032385536

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) = 4*a(n-2) for n > 31.

G.f.: (12*x^31 + 4*x^30 + x^29 + 3*x^28 + 3*x^27 + 3*x^26 + 3*x^25 + 3*x^24 + 3*x^23 + 3*x^22 + 3*x^21 + 3*x^20 + 3*x^19 + 3*x^18 + 3*x^17 + 3*x^16 + 3*x^15 + 3*x^14 + 3*x^13 + 3*x^12 + 55*x^11 + 15*x^10 - 2*x^9 + x^7 + 3*x^6 - x^5 - x^4 - 2*x^3 - 3*x - 1)/(4*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[1, i]], 2], {i, 1, stages - 1}]

Crossrefs

Cf. A280413, A280414, A280416.

Sequence in context: A277933 A278953 A280608 * A281737 A281108 A281310

Adjacent sequences: A280412 A280413 A280414 * A280416 A280417 A280418

Keyword

nonn,easy

Author

Robert Price, Jan 02 2017

Status

approved