A283752 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 758", based on the 5-celled von Neumann neighborhood.

1, 3, 5, 13, 21, 53, 93, 205, 381, 893, 1533, 3325, 6141, 14333, 24573, 53245, 98301, 229373, 393213, 851965, 1572861, 3670013, 6291453, 13631485, 25165821, 58720253, 100663293, 218103805, 402653181, 939524093, 1610612733, 3489660925, 6442450941, 15032385533

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, Dec 10 2017: (Start)

G.f.: (1 + 2*x^2)*(1 + 2*x + 4*x^3 - 8*x^4 - 8*x^5 + 24*x^6) / ((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>4.

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 758; 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. A283712, A283713, A283905.

Sequence in context: A284425 A284546 A283646 * A283602 A283817 A340400

Adjacent sequences: A283749 A283750 A283751 * A283753 A283754 A283755

Keyword

nonn,easy

Author

Robert Price, Mar 17 2017

Status

approved