A287199 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 259", based on the 5-celled von Neumann neighborhood.

1, 3, 4, 3, 16, 15, 64, 63, 256, 255, 1024, 1023, 4096, 4095, 16384, 16383, 65536, 65535, 262144, 262143, 1048576, 1048575, 4194304, 4194303, 16777216, 16777215, 67108864, 67108863, 268435456, 268435455, 1073741824, 1073741823, 4294967296, 4294967295

Offset

0,2

Comments

Initialized with a single black (ON) cell at stage zero.

Appears to differ from A277800 only at a(1). - R. J. Mathar, May 25 2017

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, May 25 2017: (Start)

G.f.: (1 + 3*x - x^2 - 12*x^3 + 12*x^5) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).

a(n) = 2^n for n>1 and even.

a(n) = 2^(n-1) - 1 for n odd.

a(n) = 5*a(n-2) - 4*a(n-4) for n>3.

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 259; 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. A287194, A287196, A287197.

Sequence in context: A172990 A084252 A342161 * A332830 A288364 A287955

Adjacent sequences: A287196 A287197 A287198 * A287200 A287201 A287202

Keyword

nonn,easy

Author

Robert Price, May 21 2017

Status

approved