A285543 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 35", based on the 5-celled von Neumann neighborhood.

1, 3, 6, 15, 28, 63, 120, 255, 496, 1023, 2016, 4095, 8128, 16383, 32640, 65535, 130816, 262143, 523776, 1048575, 2096128, 4194303, 8386560, 16777215, 33550336, 67108863, 134209536, 268435455, 536854528, 1073741823, 2147450880, 4294967295, 8589869056

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 + x - 3*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 - 2*x^2)).

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

a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3) - 2*a(n-4) + 4*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, A285542.

Sequence in context: A103529 A185378 A285563 * A318396 A034953 A086737

Adjacent sequences: A285540 A285541 A285542 * A285544 A285545 A285546

Keyword

nonn,easy

Author

Robert Price, Apr 20 2017

Status

approved