A285542 Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 35", based on the 5-celled von Neumann neighborhood.

1, 3, 3, 15, 7, 63, 15, 255, 31, 1023, 63, 4095, 127, 16383, 255, 65535, 511, 262143, 1023, 1048575, 2047, 4194303, 4095, 16777215, 8191, 67108863, 16383, 268435455, 32767, 1073741823, 65535, 4294967295, 131071, 17179869183, 262143, 68719476735, 524287

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, Apr 21 2017: (Start)

G.f.: (1 + 2*x - 6*x^2) / ((1 - x)*(1 - 2*x)*(1 + 2*x)*(1 - 2*x^2)).

a(n) = (-1 - (-2)^n + 2^n + 2^(n/2)*(1+(-1)^n)).

a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 8*a(n-4) + 8*a(n-5) for n>4.

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 35; 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. A285540, A285541, A285543.

Sequence in context: A285822 A285834 A285562 * A367774 A160612 A275324

Adjacent sequences: A285539 A285540 A285541 * A285543 A285544 A285545

Keyword

nonn,easy

Author

Robert Price, Apr 20 2017

Status

approved