A284481 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 950", based on the 5-celled von Neumann neighborhood.

1, 3, 5, 15, 23, 63, 95, 239, 383, 1023, 1535, 3839, 6143, 16383, 24575, 61439, 98303, 262143, 393215, 983039, 1572863, 4194303, 6291455, 15728639, 25165823, 67108863, 100663295, 251658239, 402653183, 1073741823, 1610612735, 4026531839, 6442450943

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, Mar 28 2017: (Start)

G.f.: (1 + 2*x + 2*x^2 + 10*x^3 - 8*x^4 + 8*x^5 - 16*x^7 + 16*x^8) / ((1 - x)*(1 - 2*x)*(1 + 2*x)*(1 + 4*x^2)).

a(n) = a(n-1) + 16*a(n-4) - 16*a(n-5) for n>5.

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 950; 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. A284479, A284480, A284482.

Sequence in context: A283908 A284409 A006977 * A290662 A003549 A018404

Adjacent sequences: A284478 A284479 A284480 * A284482 A284483 A284484

Keyword

nonn,easy

Author

Robert Price, Mar 27 2017

Status

approved